Parabolically induced unitary representations of the universal group U(F)^+ are C_0

By employing a new strategy we prove that all parabolically induced unitary representations of the Burger-Mozes universal group U(F)^+, with F being primitive, have all their matrix coefficients vanishing at infinity. This generalizes the same well-known result for the universal group U(F)^+, when F is 2-transitive.Comment: To appear in Math. Scan

A unified proof of the Howe-Moore property

We provide a unified proof of all known examples of locally compact groups that enjoy the Howe-Moore property, namely, the vanishing at infinity of all matrix coefficients of the group unitary representations that are without non-zero invariant vectors. These examples are: connected, non-compact, simple real Lie groups with finite center, isotropic simple algebraic groups over non Archimedean local fields and closed, topologically simple subgroups of Aut(T) that act 2-transitively on the boundary at infinity of T, where T is a bi-regular tree with valence > 2 at every vertex.Comment: Final version, to appear in Journal of Lie Theor

Principal subspaces of higher-level standard sl(3)^-modules

We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for \hat{\goth{sl}(3)}. As a consequence we establish new formulas for the graded dimensions of the principal subspaces corresponding to the highest-weights i\Lambda_1+(k-i)\Lambda_2, where 1 \leq i \leq k and \Lambda_1 and \Lambda_2 are fundamental weights of \hat{\goth{sl}(3)}.Comment: 25 pages; v2: minor revision

Linear Time Logics - A Coalgebraic Perspective

We describe a general approach to deriving linear time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching behaviour and linear behaviour. Concretely, we define logics whose syntax is determined by the choice of linear behaviour and whose domain of truth values is determined by the choice of branching, and we provide two equivalent semantics for them: a step-wise semantics amenable to automata-based verification, and a path-based semantics akin to those of standard linear time logics. We also provide a semantic characterisation of the associated notion of logical equivalence, and relate it to previously-defined maximal trace semantics for such systems. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear time property. We conclude with a generalisation of the logics, dual in spirit to logics with discounting, which increases their practical appeal in the context of resource-aware computation by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the results in the previous version, with new proofs; Section 6 contains new result

A hybrid model for capturing implicit spatial knowledge

This paper proposes a machine learning-based approach for capturing rules embedded in users’ movement paths while navigating in Virtual Environments (VEs). It is argued that this methodology and the set of navigational rules which it provides should be regarded as a starting point for designing adaptive VEs able to provide navigation support. This is a major contribution of this work, given that the up-to-date adaptivity for navigable VEs has been primarily delivered through the manipulation of navigational cues with little reference to the user model of navigation

A modular approach to defining and characterising notions of simulation

We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model state-based systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalgebras. We show that the expressiveness of an inductively defined logic for coalgebras w.r.t. a notion of simulation follows from an expressivity condition involving one step in the definition of the logic, and the relator inducing that notion of simulation. Moreover, we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types, to a relational level as well as to a logical level. We use these results to obtain Baltag’s logic for coalgebraic simulation, as well as notions of simulation and associated logics for a large class of non-deterministic and probabilistic systems

The flat closing problem for buildings

Using the notion of a strongly regular hyperbolic automorphism of a locally finite Euclidean building, we prove that any (not necessarily discrete) closed, co-compact subgroup of the type-preserving automorphisms group of a locally finite general non-spherical building contains a compact-by-Z^d subgroup, where d is the dimension of a maximal flat.Comment: To appear in Algebraic and Geometric Topology, final versio