Model Checking Spatial Logics for Closure Spaces

Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of programs; however, properties of space are typically not taken into account explicitly. We present a topology-based approach to formal verification of spatial properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. We lift the topological definitions to the more general setting of closure spaces, also encompassing discrete, graph-based structures. We extend the framework with a spatial surrounded operator, a propagation operator and with some collective operators. The latter are interpreted over arbitrary sets of points instead of individual points in space. We define efficient model checking procedures, both for the individual and the collective spatial fragments of the logic and provide a proof-of-concept tool

Topics in uniform continuity

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous

Rough index theory on spaces of polynomial growth and contractibility

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get higher-codimensional index obstructions to metrics of positive scalar curvature on closed manifolds with virtually nilpotent fundamental groups. We will give a computation of the homology of (a dense, smooth subalgebra of) the uniform Roe algebra of manifolds of polynomial volume growth.Comment: v4: final version, to appear in J. Noncommut. Geom. v3: added a computation of the homology of (a smooth subalgebra of) the uniform Roe algebra. v2: added as corollaries to the main theorem the multi-partitioned manifold index theorem and the higher-codimensional index obstructions against psc-metrics, added a proof of the strong Novikov conjecture for virtually nilpotent groups, changed the titl

Infinite networks and variation of conductance functions in discrete Laplacians

For a given infinite connected graph $G=(V,E)$ and an arbitrary but fixed conductance function $c$, we study an associated graph Laplacian $\Delta_{c}$; it is a generalized difference operator where the differences are measured across the edges $E$ in $G$; and the conductance function $c$ represents the corresponding coefficients. The graph Laplacian (a key tool in the study of infinite networks) acts in an energy Hilbert space $\mathscr{H}_{E}$ computed from $c$. Using a certain Parseval frame, we study the spectral theoretic properties of graph Laplacians. In fact, for fixed $c$, there are two versions of the graph Laplacian, one defined naturally in the $l^{2}$ space of $V$, and the other in $\mathscr{H}_{E}$. The first is automatically selfadjoint, but the second involves a Krein extension. We prove that, as sets, the two spectra are the same, aside from the point 0. The point zero may be in the spectrum of the second, but not the first. We further study the fine structure of the respective spectra as the conductance function varies; showing now how the spectrum changes subject to variations in the function $c$.Comment: 32 pages, 3 figure

On large indecomposable Banach spaces

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular)