27,651 research outputs found
Euler Diagram Transformations
Euler diagrams are a visual language which are used for purposes such as the presentation of set-based data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user's mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems.
We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a correspondence between manipulations of diagrams at an abstract level (such as logical reasoning steps, or simply an update of information) and the manipulation at a concrete level. Thus we facilitate the presentation of diagram changes in a manner that preserves the mental map. The approach will facilitate the realisation of reasoning systems at the concrete level; this has the potential to provide diagrammatic reasoning systems that are inherently different from symbolic logics due to natural geometric constraints. We provide a particular concrete transformation system which
preserves the important criteria of planarity and connectivity, which may form part of a framework encompassing multiple concrete systems each adhering to different sets of wellformedness conditions
Euler diagram transformations
Euler diagrams are a visual language which are used for purposes such as the presentation of set-based data or as the basis of visual logical languages which can be utilised for software specification and reasoning. Such Euler diagram reasoning systems tend to be defined at an abstract level, and the concrete level is simply a visualisation of an abstract model, thereby capturing some subset of the usual boolean logic. The visualisation process tends to be divorced from the data transformation process thereby affecting the user’s mental map and reducing the effectiveness of the diagrammatic notation. Furthermore, geometric and topological constraints, called wellformedness conditions, are often placed on the concrete diagrams to try to reduce human comprehension errors, and the effects of these conditions are not modelled in these systems. We view Euler diagrams as a type of graph, where the faces that are present are the key features that convey information and we provide transformations at the dual graph level that correspond to transformations of Euler diagrams, both in terms of editing moves and logical reasoning moves. This original approach gives a corre
The converse problem for the multipotentialisation of evolution equations and systems
We propose a method to identify and classify evolution equations and systems
that can be multipotentialised in given target equations or target systems. We
refer to this as the {\it converse problem}. Although we mainly study a method
for -dimensional equations/system, we do also propose an extension of
the methodology to higher-dimensional evolution equations. An important point
is that the proposed converse method allows one to identify certain types of
auto-B\"acklund transformations for the equations/systems. In this respect we
define the {\it triangular-auto-B\"acklund transformation} and derive its
connections to the converse problem. Several explicit examples are given. In
particular we investigate a class of linearisable third-order evolution
equations, a fifth-order symmetry-integrable evolution equation as well as
linearisable systems.Comment: 31 Pages, 7 diagrams, submitted for consideratio
A tree of linearisable second-order evolution equations by generalised hodograph transformations
We present a list of (1+1)-dimensional second-order evolution equations all
connected via a proposed generalised hodograph transformation, resulting in a
tree of equations transformable to the linear second-order autonomous evolution
equation. The list includes autonomous and nonautonomous equations.Comment: arXiv version is already officia
On the periodicity of Coxeter transformations and the non-negativity of their Euler forms
We show that for piecewise hereditary algebras, the periodicity of the
Coxeter transformation implies the non-negativity of the Euler form. Contrary
to previous assumptions, the condition of piecewise heredity cannot be omitted,
even for triangular algebras, as demonstrated by incidence algebras of posets.
We also give a simple, direct proof, that certain products of reflections,
defined for any square matrix A with 2 on its main diagonal, and in particular
the Coxeter transformation corresponding to a generalized Cartan matrix, can be
expressed as , where A_{+}, A_{-} are closely associated
with the upper and lower triangular parts of A.Comment: 12 pages, (v2) revision, to appear in Linear Algebra and its
Application
Symmetry reduction of discrete Lagrangian mechanics on Lie groups
For a discrete mechanical system on a Lie group determined by a (reduced)
Lagrangian we define a Poisson structure via the pull-back of the
Lie-Poisson structure on the dual of the Lie algebra by the
corresponding Legendre transform. The main result shown in this paper is that
this structure coincides with the reduction under the symmetry group of the
canonical discrete Lagrange 2-form on . Its
symplectic leaves then become dynamically invariant manifolds for the reduced
discrete system. Links between our approach and that of groupoids and
algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid
body is discussed as an example
- …