Convex Invariants Research Articles

Convex invariants of M-fuzzifying convex spaces

The numbers of Helly, Carathéodory and Radon are a central theme in classic convex spaces and each of them is invariant under isomorphism. The theory of M-fuzzifying convex spaces is an active research field since it is presented. This paper focus on convex invariants of M-fuzzifying convex spaces. The degrees of Helly independent, Carathéodory independent and Radon independent of non-empty set are defined in the framework of M-fuzzifying convex spaces. By those definitions, we introduce the Helly number, Carathéodory number and Radon number of M-fuzzifying convex spaces. Finally, we inspect M-fuzzifying topology and M-fuzzifying JHC which are characterized by convex invariants of M-fuzzifying convex spaces.

Representing hybrid automata by action language modulo theories

AbstractBoth hybrid automata and action languages are formalisms for describing the evolution of dynamic systems. This paper establishes a formal relationship between them. We show how to succinctly represent hybrid automata in an action language which in turn is defined as a high-level notation for answer set programming modulo theories—an extension of answer set programs to the first-order level similar to the way satisfiability modulo theories (SMT) extends propositional satisfiability (SAT). We first show how to represent linear hybrid automata with convex invariants by an action language modulo theories. A further translation into SMT allows for computing them using SMT solvers that support arithmetic over reals. Next, we extend the representation to the general class of non-linear hybrid automata allowing even non-convex invariants. We represent them by an action language modulo ordinary differential equations, which can be compiled into satisfiability modulo ordinary differential equations. We present a prototype systemcplus2aspmtbased on these translations, which allows for a succinct representation of hybrid transition systems that can be computed effectively by the state-of-the-art SMT solverdReal.

Helly and exchange numbers of geodesic and Steiner convexities in lexicographic product of graphs

We discuss the convexity invariants, namely, the exchange and Helly numbers of the Steiner and geodesic convexity in lexicographic product of graphs. We use the structure of both the Steiner and geodesic convex sets in the lexicographic product for proving the results. Along the way the exchange number of the induced path convexity in arbitrary graphs is also determined.

Regularization of non-convex strain energy function for non-monotonic stress–strain relation in the Hencky elastic–plastic model

The boundary value problem of small static deformation range is formulated as the variational problem for the displacement. The general local conditions of convexity and rank-one-convexity are obtained for smooth elastic–plastic potentials depending on two first convex invariants of the Cauchy strain tensor. It is demonstrated that all types of convexity are equivalent for the Hencky elastic–plastic potential, but, for example, the strain energy function of continuum fracture can have every type of convexity for different relationships between material parameters. The classical regularization method is applied for the construction of the lower convex envelope of the Hencky strain energy potential for the experimental non-monotonic stress–strain relation with falling part. For this relation, the concept of the Maxwell line is used by analogy with the Van der Waals’ gas theory. The results of 1-D and 2-D numerical examples show that this regularization is principally necessary for the convergence of the finite-difference methods that are usually used for numerical computation within the framework of the small deformation models with non-monotonic stress–strain relations.

Properties of normalized radial visualizations

This paper defines a class of normalized radial visualizations (NRVs) that includes the RadViz mapping onto the two-dimensional unit disk. An NRV maps high-dimensional records into lower dimensional space, where records’ images are convex combinations of the dimensions (called dimensional anchors) laid out in two dimensions as labels on a circle and in higher dimensions on the surface of a hypersphere. As radial visualizations have evolved, conjectures have been proposed for invariants, such as lines mapping to lines, and convex sets to convex sets. Some have been informally proven for RadViz. We formally establish these properties for all NRVs and illustrate them using RadViz. An extensive theory of Parallel Coordinates has been developed elsewhere with great benefit to the visualization community. Our theory should provide similar benefits for radial visualization users. We show that an NRV is the composition of a perspective and an affine transformation. This projective transformation characterization leads to a number of properties including line, point ordering and convexity invariance. Knowledge of these properties suggests that the visual existence of structure in the data can guide a visualization researcher in further productive exploration of the data. We show the established properties hold regardless of whether or not the dimensional anchors lie on the circle or the hypersphere. These insights also suggest directions for future NRV work, such as rotational preprocessing to separate data in RadViz and NRVs for better cluster visualization.

Convexity Invariance of Fuzzy Sets under the Extension Principles

We discuss the convexity invariance of fuzzy sets under the extension principles. Particularly, we give a necessary and sufficient condition for a mapping to be an inverse *-convex transformation, and also obtain some sufficient conditions for a mapping to be an *-convex transformation. Two applications are given to illustrate the obtained results. Finally, we give some applications of the main results to the hyperstructure convexity invariance of type 2 fuzzy sets under hyperalgebra operations, and to the convexity invariance of fuzzy numbers under basic arithmetic operations.

Convex independence and the structure of clone-free multipartite tournaments

We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specically , we explore the relationship between the Helly number, Radon number and rank of such digraphs. The main result is a structural theorem that describes the arc relationships among certain vertices associated with vertices of a given convexly independent set. We use this to prove that the Helly number, Radon number, and rank coincide in any clone-free bipartite tournament. We then study the relationship between Helly independence and Radon independence in clone-free multipartite tournaments. We show that if the rank is at least 4 or the Helly number is at least 3, then the Helly number and the Radon number are equal.

Turán theorems and convexity invariants for directed graphs

This paper is motivated by the desire to evaluate certain classical convexity invariants (specifically, the Helly and Radon numbers) in the context of transitive closure of arcs in the complete digraph. To do so, it is necessary to establish several new Turán type results for digraphs and characterize the associated extremal digraphs. In the case of the Radon number, we establish the following analogue for transitive closure in digraphs of Radon's classical convexity theorem [J. Radon, Mengen konvexer Körper, die einer gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113–115]: in a complete digraph on n ⩾ 7 vertices with > n 2 / 4 arcs, it is possible to partition the arc set into two subsets whose transitive closures have an arc in common.

Convexities related to path properties on graphs

A feasible family of paths in a connected graph G is a family that contains at least one path between any pair of vertices in G. Any feasible path family defines a convexity on G. Well-known instances are: the geodesics, the induced paths, and all paths. We propose a more general approach for such ‘path properties’. We survey a number of results from this perspective, and present a number of new results. We focus on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Carathéodory, Helly and Radon numbers in relation with graph invariants, such as the clique number and other graph properties.

On triangle path convexity in graphs

Convexity invariants like Caratheodory, Helly and Radon numbers are computed for triangle path convexity in graphs. Unlike minimal path convexities, the Helly and Radon numbers behave almost uniformly for triangle path convexity.

Asymptotic expansion and geometric properties of the spline collocation periodic solution of an ODE system

The asymptotic expansion of the cubic spline collocation periodic solution of ODEs is obtained, and then some criteria and techniques are developed to prove geometric properties of numerical solutions such as convexity invariance and the one-side approximation.