Showing papers on "Metric (mathematics) published in 1987"

Landscape patterns in a disturbed environment

Abstract: Deciduous forest patterns were evaluated, using fractal analysis, in the U. S. Geological Survey 1: 250,000 Natchez Quadrangle, a region that has experienced relatively recent conversion of forest cover to cropland. A perimeter-area method was used to determine the fractal dimension; the results show a different dimension for small compared with large forest patches. This result is probably related to differences in the scale of human versus natural processes that affect this particular forest pattern. By identifying transition zones in the scale at which landscape patterns change this technique shows promise for use in developing hypotheses related to scale-dependent processes and as a simple metric to evaluate changes on the earth's surface using remotely sensed data.

New lower bound techniques for robot motion planning problems

Abstract: We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the first provably intractable problem in robotics.

Congruence, similarity, and symmetries of geometric objects

Abstract: We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point setA exactly or approximately into a point setB. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).

Pinching constants for hyperbolic manifolds

Abstract: We show in this paper that for everyn≧4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with constant curvatureK≡−1. We also estimate the (pinching) constantsH for which our manifoldsV admit metrics with −1≧K≧−H.

Operator Ordering and the Flatness of the Universe

Abstract: In order to make predictions in quantum cosmology one has to resolve the problem of the operator ordering in the hamiltonian. This is equivalent to defining a differential operator on superspace, the infinite dimensional manifold of all 3-metrics and matter field configurations on a 3-surface S. We propose that this operator should be the laplacian in a natural metric. There remains a difficulty however because the metric connection on superspace depends on the metric in a nonlinear manner. This nonlinearity would be inconsistent with the interpretation of the lapse function as a Lagrange multiplier. It may cancel out if an equal number of fermion degrees of freedom are included. The probability measure on superspace would be the measure associated with this metric. In situations in which the wave function can be interpreted in terms of classical solutions by the WKB approximation, this choice of measure implies that the probability of finding the 3-metric and matter field configuration in a given region of superspace is proportional to the proper time that the solutions spend in that region. As an application we compute the probability distribution of the density parameter Ω in a minisuperspace model which is in the quantum state defined by a path integral over compact 4-geometries. If we restrict attention to a fixed value of the density, we find that the probability distribution is entirely concentrated at Ω = 1 .

Adaptive grid generation

Abstract: The fundamental principles of adaptive grid generation for the numerical analysis of physical phenomena described by systems of partial differential equations are examined in an analytical review. Topics addressed include weight functions, equidistribution in one dimension, the specification of coefficients in the linear weight, the attraction to a given grid on a curve, evolutionary forces, and metric notation. Consideration is given to curve-by-curve methods, finite-volume methods, variational methods, and temporal aspects.

An Empirical Study of Software Metrics

Abstract: Software metrics are computed for the purpose of evaluating certain characteristics of the software developed. A Fortran static source code analyzer, FORTRANAL, was developed to study 31 metrics, including a new hybrid metric introduced in this paper, and applied to a database of 255 programs, all of which were student assignments. Comparisons among these metrics are performed. Their cross-correlation confirms the internal consistency of some of these metrics which belong to the same class. To remedy the incompleteness of most of these metrics, the proposed metric incorporates context sensitivity to structural attributes extracted from a flow graph. It is also concluded that many volume metrics have similar performance while some control metrics surprisingly correlate well with typical volume metrics in the test samples used. A flexible class of hybrid metric can incorporate both volume and control attributes in assessing software complexity.

Quasi Uniformities: Reconciling Domains with Metric Spaces

Abstract: We show that quasi-metric or quasi-uniform spaces provide, inter alia, a common generalization of cpo's and metric spaces as used in denotational semantics. To accommodate the examples suggested by computer science, a reworking of basic notions involving limits and completeness is found to be necessary. Specific results include general fixed point theorem and a sequential completion construction.

Gated sets in metric spaces

Abstract: The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.

Simplified Voronoi diagrams

Abstract: We are interested in Voronoi diagrams as a tool in robot path planning, where the search for a path in an t dimensional space may be simplified to a search on an t - 1 dimensional Voronoi diagram. We define a Voronoi diagram V based on a measure of distance which is not a true metric. This formulation has lower algebraic complexity than the usual definition, which is a considerable advantage in motion planning problems with many degrees of freedom. In its simplest form, the measure of distance between a point and a polytope is the maximum of the distances of the point from the half-spaces which pass through faces of the polytope. More generally, the measure is defined in configuration spaces which represent rotation. The Voronoi diagram defined using this distance measure is no longer a strong deformation retract of free space, but it has the following useful property: any path through free space which starts and ends on the diagram can be continuously deformed so that it lies entirely on the diagram. Thus it is still complete for motion planning, but it has lower algebraic complexity than a diagram based on the Euclidean metric.

Spectral properties of generic dynamical systems

Abstract: Dynamical systems with new spectral properties are constructed using approximation theory. It is proved that these properties are generic (in a metric and topological sense) and realized within the class of smooth systems preserving a smooth measure. Bibliography: 21 titles.

Introduction to the Analysis of Metric Spaces

Abstract: Preface Part I. Metric Spaces and Normed Linear Spaces: 1. Definitions and examples 2. Balls and boundedness Part II. Limit Processes: 3. Convergence and completeness 4. Cluster points and closure 5. Application: Banach's fixed point theorem Part III. Continuity: 6. Continuity in metric spaces 7. Continuous linear mappings Part IV. Compactness: 8. Sequential compactness in metric spaces 9. Continuous functions on compact metric spaces Part V. The Metric Topology: 10. The topological analysis of metric spaces Appendices Index of notation Index.

Ambiguities in Incremental Line Rastering

Abstract: In implmenting rater grahic algorithms, it is impotant to toroughly understand behavior and implicit defaults inherent in each algorithm. Design choices must balance performance with respect to drawing speed, circult count, code space, picture fidelity, system complexity, and system consistency. For example, "close" may sound appealing when describing the match of the rastered representation to a geometirc line. An implementation, however, must quantily an error metric?such as minimum normal distance between candidate raster grid points and the geometric line?and resolve "ties" in which two candidate grid points have an equal error metric. Equal error metric ambiguity can permit algorithimic selection of raster points for a line from (X0, Y0) to (X1, Y1) to differ from points selected rastering the same line back from (X1, Y1) to (X0, Y0). Modilying a rastering algorithm to provide an exactly reversibie path, though, will cause problems when lines are rastered in a context of approximating a circle with a polygon. Only by fully understanding any algorithm can designers determine whether such pel-level anomalies are worth the code space or circuit count necessary to provide explicit user resolution, or whether a fixed default must suffice. This article discusses implementation considerations relevant to selecting and customizing incremental line-drawing algorithms to cope with such anomalies as equal error metric instances, perturbation effects of clipping, interesections in raster space, EXOR interpretations for polylines, reversibility, and fractional endpoint rounding.

Minkowski-r Back-Propagation: Learning in Connectionist Models with Non-Euclidian Error Signals

Abstract: Many connectionist learning models are implemented using a gradient descent in a least squares error function of the output and teacher signal. The present model generalizes, in particular, back-propagation [1] by using Minkowski-r power metrics. For small r's a "city-block" error metric is approximated and for large r's the "maximum" or "supremum" metric is approached. while for r=2 the standard back-propagation model results. An implementation of Minkowski-r back-propagation is described, and several experiments are done which show that different values of r may be desirable for various purposes. Different r values may be appropriate for the reduction of the effects of outliers (noise), modeling the input space with more compact clusters, or modeling the statistics of a particular domain more naturally or in a way that may be more perceptually or psychologically meaningful (e.g. speech or vision).

Coordinate Free scattered Data interpolation.

Abstract: We discuss the problem of the lack of scale invariance for scattered data interpolation methods. We introduce an affine invariant metric and show how its use can lead to methods that are translation, rotation, and scale invariant.

Voronoi diagrams in the Moscow Metric

Abstract: Most of the streets of Moscow are either radii emanating from the Kremlin, or pieces of circles around it. We show that Voronoi diagrams for $n$ points based on this metric can be computed in optimal $O(n\log n)$ time and linear space. To this end, we prove a general theorem stating that bisectors of suitably separated point sets do not contain loops if, beside other properties, there are no holes in the circles of the underlying metric. Then the Voronoi diagrams can be computed within $O(n\log n)$ steps, using a divide-and-conquer algorithm. This theorem not only applies to the Moscow metric but to a large class of metrics including the symmetric convex distance functions and all composite metrics obtained by assigning the $L_1$ or the $L_2$ metric to the regions of an arbitrary planar map.

Constructing metrics with the Heine-Borel property

Abstract: A metric space (X, d) is said to be Heine-Borel if any closed and bounded subset of it is compact. We show that any locally compact and ucompact metric space can be made Heine-Borel by a suitable remetrization. Furthermore we prove that if the original metric d is complete, then this can be done so that the new Heine-Borel metric d' is locally identical to d, i.e., for every x E X there exists a neighborhood of x on which the two metrics coincide. Introduction. By the Heine-Borel (HB) property of a metric space (X, d) we mean here that every closed bounded set is compact, i.e. bounded sets are totally bounded, and we shall say d is a Heine-Borel metric.1 We investigate here how a space can fail to be Heine-Borel. To begin with we are interested in topological conditions that insure that a metrizable space X admits an HB metric d. Such a space need not be even finite dimensional. On the other hand it is obvious that any HB space is a-compact and locally compact, and we offer in ?1 our first main result, Theorem 1, as a converse to that. In ?2 we investigate when a u-compact, locally compact metric space (X, d) admits an HB metric which is locally identical to d. Note that an HB metric is complete. In Theorem 2 we construct an HB metric locally identical to a given complete metric. This is not a definitive result, for we construct in Example 1 a metric space which is not complete, yet it too admits a locally identical HB metric. On the other hand, the usual metric d on the open interval (0,1) is an example of a space which does not admit any locally identical metric which is complete (see Remark 2 in ?2), and therefore admits no locally identical HB metric. Finally, in ?3, we present two more examples and we ask: When does a space admit an HB metric which is uniformly locally identical to a given metric? Here a definitive answer is possible, given in Theorem 3. Although the situation is seemingly close to that of ?2 it is in fact simpler. To explore further how a metric space which is a-compact, locally compact, and complete can fail to be Heine-Borel we investigate a property common to all our examples of such spaces and close with a conjecture. Received by the editors June 3, 1985 and, in revised form, December 27, 1985 and May 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 54D45, 54E35, 54E50.

Qualitative kinematics in mechanisms

Abstract: Qualitative Kinematics is the qualitative analysis of the possible geometric interactions of physical objects. This thesis investigates the problem of reasoning about the kinematic interactions between parts of a 2-dimensional mechanism as a first step towards a general theory of qualitative kinematics. We introduce the concept of place vocabularies as a generative symbolic description of the possible motion of the parts of a mechanism. Place vocabularies are particularly useful to describe higher kinematic pairs such as ratchets and escapements. We examine the requirements for the representation and introduce a definition of place vocabularies that satisfies them. We show how this representation can be computed from metric data about the shapes of the parts of a mechanism and used as a basis for qualitative envisionments of its behavior. In particular, we give implemented algorithms for computing place vocabularies for 2-dimensional higher kinematic pairs. The objects involved in the pair can have arbitrary shapes, but their boundaries must consist of straight lines and arcs, and each object must have one degree of freedom only. Complete mechanisms are analyzed as compositions of kinematic pairs. The algorithms for computing place vocabularies fall within the qualitative reasoning paradigm. They are based on splitting the computation into 2 parts: a purely symbolic reasoning part and a metric diagram. The metric diagram is an abstract device that gives access to information about quantities by evaluating predicates on them. We propose that this division is a plausible model of human reasoning. When quantitative information is incomplete, the metric diagram defines a set of landmark values for unknown metric parameters. The resulting place vocabulary changes only at these landmark values, and an exhaustive list of all place vocabularies that can be achieved by varying the parameters is found by computation at representative points in each interval. We show how the mechanism design problem of picking suitable values for parameters can be solved by searching the list, and give an implemented example. The exhaustive list forms an ambiguous but complete prediction of the possible behavior. This proves that contrary to common belief, it is in principle possible to reason about kinematics with limited or no knowledge of the metric dimensions of the objects.

A Metric for Computing Adaptive Detail in Animated Scenes Using Object-Oriented Programming

Abstract: A spatial metric for adapting the level of detail in a modelled object to achieve a convincing degree of realism on a display is formulated and illustrated with a simple implementation. This spatial metric is designed to be of most benefit to animations of natural environments. The metric can operate with a wide range of different representations: the essential requirement being that an abstract data structure with hierarchical levels of detail is formulated. Such models are most easily implemented with procedural recursive representations, such as fractals. The actor/message passing approach to modelling animation is adopted as being most appropriate and intuitive when simulating the objects and events of the natural environment. This object-oriented programming method also gives a uniform abstract interface to differing data representations.