Intersection Representation Research Articles

Impacts of geographical knowledge, spatial ability and environmental cognition on image searches supported by GIS software

Google Earth search function was used to study the impacts of small-scale spatial ability, large-scale environmental cognition, and geographical knowledge on new technology usage. The participants were 153 junior high students from central Taiwan. Geography grades served as indicators of prior knowledge, mental rotation and abstract reasoning skills as indicators of spatial ability, and sketch maps of school neighborhoods as indicators of environmental cognition (including landmark representation, intersection representation, and frame of reference). Lastly, the authors announced the landmarks searching worksheet and asked the participants to accomplish 16 familiar and unfamiliar landmark searching tasks using Google Earth with keyword search function disabled. The result showed the strongest predictor of landmark searching performance is ‘frame of reference’ in environmental cognition, followed by ‘mental rotation’ of spatial ability, ‘landmark representation’ of environmental cognition, and geographical knowledge. Google Earth landmark searches require complex cognitive processing; therefore, our conclusion is that GIS-supported image search activities give students good practice of active knowledge construction.

Schubert calculus and representations of the general linear group

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluationglN[t]\mathfrak {gl}_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection.In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.

Tensor product characterizations of mixed intersections of non quasianalytic classes and kernel theorems

AbstractMixed intersections of non quasi‐analytic classes have been studied in [12]. Here we obtain tensor product representations of these spaces that lead to kernel theorems as well as to tensor product representations of intersections of non quasi‐analytic classes on product of open or of compact sets (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On set intersection representations of graphs

AbstractThe intersection dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets Ax⊆{1, …, t} of labels to vertices x so that any two vertices x and y from different parts are adjacent if and only if |Ax∩Ay|∈L. The weight of such a representation is the sum Σx|Ax| over all vertices x. We exhibit explicit bipartite n × n graphs whose intersection dimension is (i) at least n1/|L| with respect to any type L, (ii) at least $\sqrt{n}$ with respect to any type of the form L={k, k+ 1, …}, and (iii) at least n1/|R| with respect to any type of the form L={k|k modp∈R}, where p is a prime number. We also show that any intersection representation of a Hadamard graph must have weight about n lnn/ln lnn, independent on the used type L. Finally, we formulate several problems about intersection dimensions of graphs related to some basic open problems in the complexity of boolean functions. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 55‐75, 2009

On economical set representations of graphs

Graphs and Algorithms In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering.

Formula omitted]-interval-filament graphs

Gavril [F. Gavril, Maximum weight independent sets and cliques in intersection graphs of filaments, Inform. Process. Lett. 73 (2000) 181–188] defined two new families of intersection graphs: the interval-filament graphs and the subtree-filament graphs. The complements of interval-filament graphs are the cointerval mixed graphs and the complements of subtree-filament graphs are the cochordal mixed graphs. The family of interval-filament graphs contains the families of cocomparability, polygon-circle, circle and chordal graphs. In the present paper we introduce a generalization of the subtree-filament graphs, namely, the 3 D -interval-filament graphs. We prove that the family of 3D-interval-filament graphs, the family of complements of co( interval-filament) mixed graphs and the family of overlap graphs of interval-filaments are the same, and show that every 3 D -interval-filament graph has an intersection representation by a family of piecewise linear filaments. We use the properties of these graphs to describe an algorithm for maximum weight holes of a given parity in 3 D -interval-filament graphs and an algorithm for antiholes of a given parity in interval-filament and subtree-filament graphs. We define various subfamilies of 3 D -interval-filament graphs and characterize them as overlap graphs and as complements of H -mixed graphs.

Intersection representations of matrices by subtrees and unicycles on graphs

Consider a 0 – 1 matrix M ( i , j ) with columns C = { c 1 , c 2 , … , c m } , and rows R, or—equivalently—a hypergraph M ( R , C ) having M as its adjacency matrix (where R are the vertices and C are the hyperedges). Denote r i = { c j | c j ∈ C and M ( i , j ) = 1 } . We consider the following two problems: (a) Is there a graph H, with vertex set C, such that every vertex subgraph H ( r i ) of H is a tree and the intersection of every two such trees is also a tree? (b) Is there a graph H, with vertex set C, such that every H ( r i ) is a unicycle and the intersection of every two and every three unicycles is a tree? These questions occur in application areas such as database management systems and computational biology; e.g., in the latter they arise in the context of the analysis of biological networks, primarily for the purpose of data clustering. We describe algorithms to find such intersection representations of a matrix M (and equivalently of the hypergraph M), when they exist.

Complexity results on graphs with few cliques

Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

Characteristics and structures of weak efficient surfaces of production possibility sets

This paper studies the characteristics and structure of the weak surface of the production possibility set. We apply techniques and methods of transferring a polyhedral cone from its intersection form to its sum form, identify an intersection representation of the production possibility set. We give the structure theorem of weak surface of the production possibility set, which includes three complementary slackness conditions. We define the input weak efficiency and output weak efficiency for different DEA models according to the representation of the intersection form. It investigates the characteristics of the weak surfaces, and proves the structure theorems of input weak DEA efficiency and output weak DEA efficiency. The structure theorems establish weighted combination of inputs and outputs that are weak DEA efficient. Numerical examples are provided for illustration.

Calibration and Application of a Simulation-Based Dynamic Traffic Assignment Model

The calibration and application of a simulation-based dynamic traffic assignment (DTA) model on a portion of the city of Calgary road network in Alberta, Canada, are discussed. The DTA model iteratively reassigns flow to paths by using the method of successive averages on the basis of travel times obtained with a traffic simulation model. The original subnetwork extracted from a regional planning model was enriched by a great increase in the number of zones. The DTA origin-destination matrix was estimated from an extensive database of turning movement counts via a trip generation/distribution model and a matrix adjustment algorithm. The network topology was enhanced by the addition of an interchange and a more precise representation of arterial intersections, including traffic signal control plans. A set of 1-h turning counts was used to calibrate the DTA model by adjusting local parameters such as gap-acceptance values, as well as global parameters such as average vehicle length. The final model results were compared with an independent set of 15-min turning movement counts. The resulting R2 values, which ranged from .91 to .96, lead to a high degree of confidence in the model results.

Linear perturbation methods for topologically consistent representations of free-form surface intersections

By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact agreement between the images in R 3 of parameter-domain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed. This exact agreement of contiguous trimmed surfaces is essential to achieving topologically consistent solid model constructions through Boolean operations, and has a profound impact on the efficiency and reliability of applications such as meshing, rendering, and computing volumetric properties. Moreover, the control point perturbations require only the solution of a linear system for their determination. The basic principles of this approach to topologically consistent surface trimming curves are described, and example results from the implementation of a simple instance of the method are presented.

A formal study of linearity axioms for fuzzy orderings

This contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings. Different existing concepts are evaluated with respect to three fundamental correspondences from the classical case—linearizability of partial orderings, intersection representation, and one-to-one correspondence between linearity and maximality. As a main result, we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case. If we do not require a one-to-one correspondence between linearity and maximality, however, we obtain that an implication-based definition appears to constitute a sound compromise, in particular, if Łukasiewicz-type logics are considered.

Intersection representation of digraphs in trees with few leaves

The leafage of a digraph is the minimum number of leaves in a host tree in which it has a subtree intersection representation. We discuss bounds on the leafage in terms of other parameters (including Ferrers dimension), obtaining a string of sharp inequalities. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 340–353, 1999

Simultaneous intersection representation of pairs of graphs

Simultaneous intersection representation of pairs of graphs

Simultaneous intersection representation of pairs of graphs

We characterize the pairs (G1, G2) of graphs on a shared vertex set that are intersection polysemic: those for which the vertices may be assigned subsets of a universal set such that G1 is the intersection graph of the subsets and G2 is the intersection graph of their complements. We also consider several special cases and explore bounds on the size of the universal set. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 171–190, 1999

On the Fractional Intersection Number of a Graph

An intersection representation of a graph G is a function f : VOGU!2 S (where S is any set) with the property that uv A EOGU if and only if fOuUV fOvU0 h. The size of the representation is jSj. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that program gives the fractional intersection number. This is in consonance with fractional versions of other graph invariants such as matching number, chromatic number, edge chromatic number, etc. We examine cases where the fractional and ordinary intersection numbers are the same (interval and chordal graphs), as well as cases where they are wildly dierent (complete multipartite graphs). We find the fractional intersection number of almost all graphs by considering random graphs.

Dynamic conflict-free routing of automated guided vehicles

This paper presents work performed to address design and operational control issues for an Automated Guided Vehicle (AGV) based material handling system. Various design issues connected with flow path design, such as traffic flow along the guide paths, location and representation of intersections, buffers, pickup and dropoff points, are outlined. A network representation of the AVG system (AGVS) is presented. Operational control factors, such as demand selection and assignment, route planning and traffic management, idle AGV positioning, and AGV characteristics, are addressed. In particular, the performance of six demand selection policies, four demand assignment policies, and two idle AGV positioning policies are outlined. A very effective route-generation technique that provides conflict-free routes for multiple AGVs with varying speeds is presented. A large-scale simulation of a dynamic batch type manufacturing system to test the feasibility of the proposed scheme for real time control of AGVs and to determine the effect of several factors on several performance measures is presented. Our findings indicate that performance is significantly affected by vehicle speed, number of vehicles, demand arrival interval, idle AGV positioning, and the demand selection and assignment policy in use.

Completeness for intersection classes

An intersection representation of a graph is a function gf mapping vertices to sets such that for any u ≠ v, u is adjacent to v iff φ( u) ∩ φ( v) ≠ ⊘. The intersection class defined by a set of sets ∑ is the set of all graphs having an intersection representation using sets from ∑. Interval graphs and chordal graphs are well-studied examples of intersection classes. This paper introduces the notion of completeness for intersection classes. That is, determining precisely what restrictions can be made on the allowable sets without losing the power to represent any graph in the intersection class. The solution to this problem is presented for the chordal graphs.

On intersection representations of co-planar graphs

We show that complements of planar graphs have intersection representations by convex sets in the plane, i.e., for every planar graph, one can assign convex sets in the plane to its vertices in such a way that two of the sets are disjoint if and only if the correspondning vertices are adjacent. This fact has a complexity consequence — it follows that the problem of determining the clique number of an intersection graph of convex sets in the plane is NP-hard. We note that the complexity of this problem for intersection graphs of straight line segments in the plane is unknown.

The leafage of a chordal graph

The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on n-vertex graphs is n − lg n− 1 2 lg lg n+O(1). The proper leafage l(G) is the minimum number of leaves when no subtree may contain another; we obtain upper and lower bounds on l(G). Leafage equals proper leafage on claw-free chordal graphs. We use asteroidal sets and structural properties of chordal graphs.