The classical random walk isomorphism theorems relate the local times of a continuous-time to square Gaussian free field. A field is spin system that takes values in Euclidean space, and this article generalises systems taking hyperbolic spherical geometries. corresponding walks are no longer Markovian: they vertex-reinforced vertex-diminished jump processes. We also investigate supersymmetric ...

Scalable sets are de ned and their properties studied. It is shown that the set of scalable sets is the isomorphism closure of the set of rankable sets and that every scalable set is P-isomorphic to some rankable set. Scalable sets coincide with P-printable sets when sparse, and with P-paddable sets when thick. Using scalability as a tool, the P-isomorphism question for polynomial-time computab...

Subgraph Isomorphism is a very basic graph problem, where given two graphs G and H one is to check whether G is a subgraph of H . Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it generalizes problems such as Clique, r-Coloring, Hamiltonicity, Set Packing and Bandwidth. However, for all of the mentioned problems 2O(n) time algorithms exist, so a n...

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L1 and L2 the goal is to decide whether there exists an orthogonal linear transformation mapping L1 to L2. Our main result is an algorithm for this problem running in time nO(n) times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, whic...

 In this paper by use of mathematical modeling of an observer [14,15] the notion of relative information functional for relative dynamical systemson compact metric spaces is presented. We extract the information function ofan ergodic dynamical system (X,T) from the relative information of T fromthe view point of observer χX, where X denotes the base space of the system.We also generalize the in...