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65
Analysis of the clustering properties of the Hilbert spacefilling curve
 IEEE Transactions on Knowledge and Data Engineering
, 2001
"... AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatiotemporal databases and image compression. In these applications, one of the most desired properties from such linear mappings is clustering, whic ..."
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Cited by 141 (10 self)
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AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatiotemporal databases and image compression. In these applications, one of the most desired properties from such linear mappings is clustering, which means the locality between objects in the multidimensional space being preserved in the linear space. It is widely believed that the Hilbert spacefilling curve achieves the best clustering [1], [14]. In this paper, we analyze the clustering property of the Hilbert spacefilling curve by deriving closedform formulas for the number of clusters in a given query region of an arbitrary shape (e.g., polygons and polyhedra). Both the asymptotic solution for the general case and the exact solution for a special case generalize previous work [14]. They agree with the empirical results that the number of clusters depends on the hypersurface area of the query region and not on its hypervolume. We also show that the Hilbert curve achieves better clustering than the z curve. From a practical point of view, the formulas given in this paper provide a simple measure that can be used to predict the required disk access behaviors and, hence, the total access time.
Analysis of projection methods for solving linear systems with multiple righthand sides
 SIAM J. Sci. Comput
, 1997
"... Abstract. We analyze a class of Krylov projection methods but mainly concentrate on a specific conjugate gradient (CG) implementation by Smith, Peterson, and Mittra [IEEE Transactions on Antennas and Propogation, 37 (1989), pp. 1490–1493] to solve the linear system AX = B, where A is symmetric posit ..."
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Cited by 34 (1 self)
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Abstract. We analyze a class of Krylov projection methods but mainly concentrate on a specific conjugate gradient (CG) implementation by Smith, Peterson, and Mittra [IEEE Transactions on Antennas and Propogation, 37 (1989), pp. 1490–1493] to solve the linear system AX = B, where A is symmetric positive definite and B is a multiple of righthand sides. This method generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems orthogonally onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated with another unsolved system as a seed until all the systems are solved. We observe in practice a superconvergence behavior of the CG process of the seed system when compared with the usual CG process. We also observe that only a small number of restarts is required to solve all the systems if the righthand sides are close to each other. These two features together make the method particularly effective. In this paper, we give theoretical proof to justify these observations. Furthermore, we combine the advantages of this method and the block CG method and propose a block extension of this single seed method.
The Garden of Knowledge as a Knowledge Manifold  A Conceptual Framework for Computer Supported Subjective Education
 CID17, TRITANAD9708, DEPARTMENT OF NUMERICAL ANALYSIS AND COMPUTING SCIENCE
, 1997
"... This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in te ..."
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Cited by 22 (14 self)
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This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols  procedures that are implemented on top of the participating subjective knowledgepatches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inversedirect limit duality of Category Theory.
Spatially balanced sampling of natural resources
 Journal of the American Statistical Association
, 2004
"... The spatial distribution of a natural resource is an important consideration in designing an ef � cient survey or monitoring program for the resource. Generally, sample sites that are spatially balanced, that is, more or less evenly dispersed over the extent of the resource, are more ef � cient than ..."
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Cited by 19 (2 self)
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The spatial distribution of a natural resource is an important consideration in designing an ef � cient survey or monitoring program for the resource. Generally, sample sites that are spatially balanced, that is, more or less evenly dispersed over the extent of the resource, are more ef � cient than simple random sampling. We review a uni � ed strategy for selecting spatially balanced probability samples of natural resources. The technique is based on creating a function that maps twodimensional space into onedimensional space, thereby de � ning an ordered spatial address. We use a restricted randomization to randomly order the addresses, so that systematic sampling along the randomly ordered linear structure results in a spatially wellbalanced random sample. Variable inclusion probability, proportional to an arbitrary positive ancillary variable, is easily accommodated. The basic technique selects points in a twodimensional continuum, but is also applicable to sampling � nite populations or onedimensional continua embedded in twodimensional space. An extension of the basic technique gives a way to order the sample points so that any set of consecutively numbered points is in itself a spatially wellbalanced sample. This latter property is extremely useful in adjusting the sample for the frame imperfections common in environmental sampling.
Probability, Random Processes, and Ergodic Properties
, 2001
"... ar expended. A more idealistic motivation was that the presentation had merit as filling a unique, albeit small, hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statis ..."
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Cited by 18 (0 self)
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ar expended. A more idealistic motivation was that the presentation had merit as filling a unique, albeit small, hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statistics department, at least not before they need some of the material in their research. In addition, many of the existing mathematical texts on the subject are hard for this audience to follow, and the emphasis is not well matched to engineering applications. A notable exception is Ash's excellent text [1], which was likely influenced by his original training as an electrical engineer. Still, even that text devotes little e#ort to ergodic theorems, perhaps the most fundamentally important family of results for applying probability theory to real problems. In addition, there are many other special topics that are given little space (or none at all) in most texts on advanced probability and ran
A General Formulation of Conceptual Spaces as a Meso Level Representation
, 2001
"... Representing cognitive processes remains one of the great research challenges. Many important application areas, such as clinical diagnosis, operate in an environment of relative magnitudes, counts, shapes, colours, etc. which are not well captured by current representational approaches. This paper ..."
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Cited by 16 (1 self)
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Representing cognitive processes remains one of the great research challenges. Many important application areas, such as clinical diagnosis, operate in an environment of relative magnitudes, counts, shapes, colours, etc. which are not well captured by current representational approaches. This paper presents conceptual spaces as a meso level representation for cognitive systems, between the high level symbolic representations and the subconceptual connectionist representations which have dominated AI. Conceptual spaces emphasize orders and measures and therefore naturally represent counts, magnitudes, and volumes. Taking Grdenfors' decadelong investigation of conceptual spaces [Grdenfors, Conceptual Spaces: The Geometry of Thought, MIT Press, 2000] as start point, the paper presents a formal foundation for conceptual spaces, shows how they are theoretically and practically linked to higher and lower representational levels, and develops dynamics which allow the orbits of states in the space to solve appropriate meso level reasoning tasks. Interpretations of conceptual spaces are given to illustrate the formal definitions and show the flexibility of the representation. 2001 Elsevier Science B.V. All rights reserved.
Convergence theorem for a general class of power control algorithms
 IEEE Transactions on Communications
, 2004
"... Abstract—We consider the convergence issues of distributed powercontrol algorithms for mobile cellular systems. A convergence theorem for powercontrol algorithms of canonical type is proved. Our result generalizes Yates ’ framework and provides a new outlook on the problem. The general applicabili ..."
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Cited by 14 (1 self)
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Abstract—We consider the convergence issues of distributed powercontrol algorithms for mobile cellular systems. A convergence theorem for powercontrol algorithms of canonical type is proved. Our result generalizes Yates ’ framework and provides a new outlook on the problem. The general applicability of the theorem is demonstrated by showing that many wellknown distributed algorithms are canonical. Furthermore, by devising some new discrete algorithms, we exemplify how the theorem can be used to aid new design. Index Terms—Canonical algorithm, distributed algorithms, framework, power control. I.
Notes on Spectral Theory
, 1966
"... Second edition revised and typeset by the Author. CONTENTS ..."
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Cited by 14 (0 self)
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Second edition revised and typeset by the Author. CONTENTS
Proofs and Pictures Proving the Diamond Lemma with the grover Theorem Proving System
, 1995
"... In this paper we describe a theorem proving system called grover. grover is novel in that it may be guided in its search for a proof by information contained in a diagram. There are two parts to the system: the underlying theorem prover, called &, and the graphical subsystem which examines the diagr ..."
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Cited by 11 (2 self)
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In this paper we describe a theorem proving system called grover. grover is novel in that it may be guided in its search for a proof by information contained in a diagram. There are two parts to the system: the underlying theorem prover, called &, and the graphical subsystem which examines the diagram and makes calls to the underlying prover on the basis of the information found there. We have used grover to prove the Diamond Lemma, a nontrivial theorem from the theory of wellfounded relations. Key words. Automated reasoning, graphical theorem proving, proof strategies. This material is based upon work supported by the National Science Foundation under award number ISI8701133. 1 INTRODUCTION 2 1 Introduction Open almost any mathematics text book and you will find, along with the familiar symbolism of mathematics and motivational text, many diagrams which are included to help the reader visualize the particular point being made. One might be tempted to conclude that mathema...
How Good Are Fuzzy IfThen Classifiers?
"...  This paper gives some known theoretical results about fuzzy rulebased classiers and oers a few new ones. The ability of TakagiSugenoKang (TSK) fuzzy classiers to match exactly and to approximate classi cation boundaries is discussed. The lemma by Klawonn and Klement about the exact match of a c ..."
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Cited by 10 (0 self)
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 This paper gives some known theoretical results about fuzzy rulebased classiers and oers a few new ones. The ability of TakagiSugenoKang (TSK) fuzzy classiers to match exactly and to approximate classi cation boundaries is discussed. The lemma by Klawonn and Klement about the exact match of a classication boundary in < 2 is extended from monotonous to arbitrary functions. Equivalence between fuzzy rulebased and nonfuzzy classiers (1nn and Parzen) is outlined. We specify the conditions under which a class of fuzzy TSK classiers turn into lookup tables. It is shown that if the rule base consists of all possible rules (all combinations of linguistic labels on the input features), the fuzzy TSK model is a lookup classier with hyperbox cells, regardless of the type (shape) of the membership functions used. The question \why fuzzy?" is addressed in the light of these results. I. Introduction Fuzzy rulebased classiers are a popular counterpart of fuzzy control systems. ...