Exact Isomorphism Research Articles

Information Systems Development and Music: The Exploration of a Parallelism

The author examines numerous parallels between information systems and music. Roles played by individuals involved in the creation and performance of music are shown to correspond to parallel roles in the creation and implementation of information systems. Examples include correspondences between the user, in information systems, and the audience, in music. Similarly, many processes or elements---such as analysis, design and programming, among others---are common to both domains. For instance, thematic transformation in music composition can be seen to correspond to a form of programming logic in information systems. The author also notes that while many elements of information systems can be mapped onto musical functions, the analogy does not hold in all instances---in particular, the importance of subjectivity and interpretation in music does not have a parallel in information systems, where the complete objectivity of the computer and reproducibility of results are paramount. The development of an information system and the creation and performance of a piece of music may be viewed as having a great deal in common. Information system roles such as user, analyst, designer, programmer and implementor can be mapped onto various music roles, such as audience, composer, conductor, performer, etc. (although not necessarily respectively). Going beyond personal roles, we can draw other analogies in regards the processes common to each domain, such as analysis, design, programming, testing, implementation and maintenance. Although the relationship is not an exact isomorphism, there are significant parallels. In exploring these parallels, I will present here the most common elements from the domain of information systems development, and for each explore the corresponding element from the realm of music.

Diameter and Treewidth in Minor-Closed Graph Families

It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.

Error-Correcting Graph Isomorphism Using Decision Trees

In this paper we present a fast algorithm for the computation of error-correcting graph isomorphisms. The new algorithm is an extension of a method for exact subgraph isomorphism detection from an input graph to a set of a priori known model graphs, which was previously developed by the authors. Similar to the original algorithm, the new method is based on the idea of creating a decision tree from the model graphs. This decision tree is compiled off-line in a preprocessing step. At run time, it is used to find all error-correcting graph isomorphisms from an input graph to any of the model graphs up to a certain degree of distortion. The main advantage of the new algorithm is that error-correcting graph isomorphism detection is guaranteed to require time that is only polynomial in terms of the size of the input graph. Furthermore, the time complexity is completely independent of the number of model graphs and the number of edges in each model graph. However, the size of the decision tree is exponential in the size of the model graphs and the degree of error. Nevertheless, practical experiments have indicated that the method can be applied to graphs containing up to 16 vertices.

Extending the dual of the Petersen graph

By hypotheses (a) and (c), if {i, j} is a 2-element subset of the type set I of Γ, then there are flags F of co-type {i, j}, such that res(F ) is a rank 2 geometry over the type set {i, j}, and all such {i, j}-residues are isomorphic. Hence, such a geometry Γ can be described by a diagram, in which the types (or even the exact isomorphism types) of all rank-2-residues are listed; this is often done in a graphical way. A graph D is drawn having vertices i ∈ I , one vertex for every type of objects in a hypothetical geometry belonging to the diagram, and the way two vertices i, j ∈ I are connected in the diagram reflects the type Dij of some rank 2 geometry.

Numerical Simulations in Superfluid Helium

In 1953, Richard Feynman showed how superfluidity can be understood from the point of view of imaginary-time path integrals. In this picture, each atom becomes a cyclic polymer and the superfluid transition occurs when the atomic paths join up to make larger units which stretch a macroscopic distance. Boson statistics imply that all possible ways of connecting or "permuting" the "polymers" have equal weight. Feynman's path integrals are an exact isomorphism between quantum statistical mechanics of bosons and the classical statistical mechanics of these special "polymers". We have implemented Feynman's path integral method on a computer. Efficient methods, based on the Metropolis algorithm have been developed to move the paths quickly through path configuration space and permutation space. In addition, accurate approximations to the many-body off-diagonal density matrix involving pairs and triplets of atoms have been calculated.

Isomorphisms for random sequential packing on lattices

Recently Nakamura (1986) reported results for two classes of random sequential packing problems on n*n square lattices. Non-overlapping squares covering a2 cells are placed on the lattice either A allowing, or B forbidding contact. Here the author demonstrates an exact isomorphism between these two classes, as well as a broader range of models (and their d-dimensional analogues). Nakamura first took the n to infinity infinite-lattice limit and then analysed convergence of the packing fraction, p (i.e. saturation or jamming coverage), to the a to infinity continuum limit (where A and B coincide). The author obtains p values for model A with n= infinity from the generalised Palasti conjecture (using exact d=1 values), and develops a corresponding a to infinity asymptotic expansion; the corresponding p values for model B are obtained from isomorphism arguments. These facilitate analysis of Nakamura's results. Isomorphisms and continuum limit behaviour are also discussed for processes on other lattices.

Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids

From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.