We review key mathematical concepts used in the theory of Statistical Shape Analysis (SSA). The treatment is elementary and aims at providing a brief guide to a large number of results and ideas which are dispersed over an also very large literature on Non-Euclidean Geometry, Differential Geometry, and Topology. The goal is to provide an introduction to the ideas SSA touches in these areas to r...

This paper addresses the problem of designing a submodule of a given system of communicating timed I/O automata. The problem may be formulated mathematically by the equation (C||X)rA under the constraint IX=In, where C represents the specification of the known part of the system, called the context, A represents the specification of the whole system, X represents the specification of the submod...

We consider equivalence relations for Fluid Stochastic Petri Nets (FSPNs). Based on equivalence relations for Stochastic Petri Nets (SPNs), which are derived from lumpability for Markov Chains, and from lumpability for certain classes of differential equations, we define an equivalence relation for FSPNs. Lumpability for the differential equations is based on a finite discretization approach an...

We study, in an abstract context, equivalence relations which are invariant under group actions. More precisely, we fix a transformation group, and we study the orbital equivalence relations (i.e. orbit equivalence relations of normal subgroups) and a wider class of weakly orbital equivalence relations. For these sorts of relations we show (under some additional assumptions) that if each class ...

We use modiied Tsirelson's spaces 3] to prove that there is no nite basis for turbulent Polish group actions. This answers a question of 1. Introduction One of the trends in Descriptive Set Theory during the last decade or so concerns the study of classiiability of Borel equivalence relations. Particularly well-studied are the orbit equivalence relations of Polish group actions (see 7, 10, 11, ...

This paper is about ‡ow-orbit equivalence, a topological analogue of even Kakutani equivalence. In addition to establishing many basic facts about this relation, we characterize the conjugacies of induced systems that can be extended to a ‡ow-orbit equivalence. We also describe the relationship between ‡ow-orbit equivalence and a distortion function of an orbit equivalence. We show that if the ...

Equivalence checking is a formal verification approach that consists in proving two programs or models are related modulo some equivalence relation, one included the other preorder relation. In case of concurrent systems, which often represented using labelled transition relations used for bisimulations and their simulation preorders. probabilistic stochastic usually Markov chains, lumpability,...

Recall that last time we have introduced the set of weights Wt(n) ⊂ C and an equivalence relation ∼ on Wt(n). Our goal is to describe this set and this equivalence relation – the equivalence classes are in a natural bijection with Irr(Sn). The first step here is as follows. Pick a path P = V 1 → V 2 → . . . → V n in the branching graph and an integer k with 1 < k < n. We fix all vertices but V ...

In this paper, we de ne the LP equivalence relation, and introduce an LP characteristic vector, which is invariant under this relation. We derive the characteristic vectors for functions on ve or fewer input variables and show that the characteristic vectors uniquely identify the equivalence classes. We have obtained the table of minimum AND-EXOR expressions (MESOPs) for the representative func...

This paper introduces a new conformance relation between a specification and an implementation of a distributed system. It is based on a local view which allows to avoid or reduce the state explosion problem. The conformance relation is defined via Petri nets and shows not only equivalence between transitions but also equivalence between local states. This equivalence depends on the structural ...