The present book provides an introduction to using space-filling curves (SFC) as tools in scientific computing. Special focus is laid on the representation of SFC and on resulting algorithms. For example, grammar-based techniques are introduced for traversals of Cartesian and octree-type meshes, and arithmetisation of SFC is explained to compute SFC mappings and indexings. The locality properties of SFC are discussed in detail, together with their importance for algorithms. Templates for parallelisation and cache-efficient algorithms are presented to reflect the most important applications of SFC in scientific computing. Special attention is also given to the interplay of adaptive mesh refinement and SFC, including the structured refinement of triangular and tetrahedral grids. For each topic, a short overview is given on the most important publications and recent research activities.
Linking the differing techniques deployed in describing space-filling curves to their corresponding algorithms, this book introduces SFCs as tools in scientific computing, focusing in particular on the representation of SFCs and on the ...
Author: Michael Bader
Publisher: Springer Science & Business Media
The subject of space-filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Working in this area is like skating on the edge of reason. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by W. Sierpiriski in 1912. At that time, the subject was still in its infancy and the most interesting and perplexing results were still to come. Besides, Sierpiriski's paper was written in Polish and published in a journal that is not readily accessible (Sierpiriski ). Most of the early literature on the subject is in French, German, and Polish, providing an additional raison d'etre for a comprehensive treatment in English. While there was, understandably, some intensive research activity on this subject around the turn of the century, contributions have, nevertheless, continued up to the present and there is no end in sight, indicating that the subject is still very much alive. The recent interest in fractals has refocused interest on space filling curves, and the study of fractals has thrown some new light on this small but venerable part of mathematics. This monograph is neither a textbook nor an encyclopedic treatment of the subject nor a historical account, but it is a little of each. While it may lend structure to a seminar or pro-seminar, or be useful as a supplement in a course on topology or mathematical analysis, it is primarily intended for self-study by the aficionados of classical analysis.
... K., 1,54, 55, 136, 140, 146, 180 biography, 55 Knopp's generation of von Koch
curve, 140 Osgood curves, 137 ff. representation of Sierpiński's spacefilling curve
, 51 ff. Knuth, D.E., 162, 165, 178 Koch, H. von, 54, 181 curve, 54, 78, 140, 146, ...
Author: Hans Sagan
Publisher: Springer Science & Business Media
Author: Richard Engler Krebs
Category: Curves, Algebraic
Author: Fracchia, F. David
Publisher: Regina : Department of Computer Science, University of Regina
Scientific Essay from the year 2015 in the subject Mathematics - Miscellaneous, language: English, abstract: Representation of two dimensional objects into one dimensional space is simple and efficient when using a two coordinate system imposed upon a grid. However, when the two dimensions are expanded far beyond visual and sometimes mental understanding, techniques are used to quantify and simplify the representation of such objects. These techniques center around spatial interpretations by means of a space-filling curve. Since the late 1800's, mathematicians and computer scientists have succeeded with algorithms that express high dimensional geometries. However, very few implementations of the algorithms beyond three dimensions for computing these geometries exist. We propose using the basic spatial computations developed by pioneers in the field like G. Peano, D. Hilbert, E. H. Moore, and others in a working model. The algorithms in this paper are fully implemented in high-level programming languages utilizing a relation database management system. We show the execution speeds of the algorithms using a space-filling curve index for searching compared to brute force searching. Finally, we contrast three space-filling curve algorithms: Moore, Hilbert, and Morton, in execution time of searching for high dimensional data in point queries and range queries.
Scientific Essay from the year 2015 in the subject Mathematics - Miscellaneous, language: English, abstract: Representation of two dimensional objects into one dimensional space is simple and efficient when using a two coordinate system ...
Author: Ankush Chauhan
Publisher: Grin Publishing
Introduction to Global Optimization Exploiting Space-Filling Curves provides an overview of classical and new results pertaining to the usage of space-filling curves in global optimization. The authors look at a family of derivative-free numerical algorithms applying space-filling curves to reduce the dimensionality of the global optimization problem; along with a number of unconventional ideas, such as adaptive strategies for estimating Lipschitz constant, balancing global and local information to accelerate the search. Convergence conditions of the described algorithms are studied in depth and theoretical considerations are illustrated through numerical examples. This work also contains a code for implementing space-filling curves that can be used for constructing new global optimization algorithms. Basic ideas from this text can be applied to a number of problems including problems with multiextremal and partially defined constraints and non-redundant parallel computations can be organized. Professors, students, researchers, engineers, and other professionals in the fields of pure mathematics, nonlinear sciences studying fractals, operations research, management science, industrial and applied mathematics, computer science, engineering, economics, and the environmental sciences will find this title useful .
This work also contains a code for implementing space-filling curves that can be used for constructing new global optimization algorithms.
Author: Yaroslav D. Sergeyev
Publisher: Springer Science & Business Media
Curves and Their Parametrizations 2 . The Areas of Images of Curves 3 . Limit
Points and Limit Curves Completeness and the Dominated Convergence
Theorem 5 . Nowhere Differentiable Curves 6 . Peano ' s Space Filling Curve
Category: Educational innovations
* A lovingly-crafted visual expedition, lead by a lifelong fractal wizard with an obsession for categorizing fractal species * Hundreds of beautiful color images * An in-depth taxonomy of Koch-constructed Fractal Curves * An intuitive introduction to Koch construction * A must-read for anyone interested in fractal geometry
* A lovingly-crafted visual expedition, lead by a lifelong fractal wizard with an obsession for categorizing fractal species * Hundreds of beautiful color images * An in-depth taxonomy of Koch-constructed Fractal Curves * An intuitive ...
Author: Jeffrey Ventrella
We will first describe the use of space-filling curves in Section 2.1, and the use of
clustering in Section 2.2. 2.1. Space-filling curves A curve is the continuous
image of a closed real interval. A space-filling curve is a curve which possesses
Category: Database management
MR2133732 ( 2006b : 65188 ) 65N50 Maubach , Joseph M . ( NL - EIND ;
Eindhoven ) Space - filling curves for 2 ... locally refined computational 2 -
meshes which can be very efficiently load - balanced with the use of a space -
filling curve .
A collection of 69 papers which were presented at the IS&T/SPIE Electronic Imaging Symposium, 1999. They appear in 13 sessions on subjects such as: image retrieval applications; multimedia management and retrieval systems; video retrieval; and image browsing.
... for selecting appropriate orderings . We will first describe the use of space -
filling curves in Section 2 . 1 , and the use of clustering in Section 2 . 2 . 2 . 1 .
Space - filling curves A curve is the continuous image of a closed real interval .
Author: Minerva Ming-Yee Yeung
Publisher: Society of Photo Optical
Category: Technology & Engineering
This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.
This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary ...
Author: Felix Hausdorff
Publisher: American Mathematical Soc.
Space - Filling Curves M . K . SINGAL Professor of Mathematics , Meerut
University , Meerut - 250 004 In this article the author had described Peano ' s
discovery of Space - Filling Curves and has given several examples of such
curves due to ...
On p-Norm Based Locality Measures of Space-Filling Curves H. K. Dai and H. C.
Su* Computer Science Department, Oklahoma State ... A discrete space-filling
curve provides a linear indexing or traversal of a multi-dimensional grid space.
Category: Computer algorithms
Run - time of space - filling curves and Metis needed to partition " hermes _ all ” .
" dime20 ” graph . All space - filling curves need much less time for their
computation than either kmetis or even the recursive partitioner pmetis . This is
Category: Parallel processing (Electronic computers)
Poster Abstract : Serial Data Aggregation Using Space - filling Curves in Wireless
Sensor Networks Swapnil Patil and Samir R . Das Dept . of Computer Science
SUNY at Stony Brook Stony Brook , NY 11790 swapnil , samir ...
Category: Embedded computer systems
template 1 . iterate 2. iterate discrete curve for adaptive grid 33 Hilbert-curve
Peano-curve stacks for which we can even completely avoid 'jumps' in the
adresses instead of only reducing their probability or frequency. 2 Space-Filling
On a value bound - plane , the combined values calculated by the space - filling
curve are clustered in an rn rectangle . Depending on the clustering behavior of
the space - filling curve , the number of contiguous disk areas required to store
Category: Data structures (Computer science)