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Review Article Survey Report on Space Filling Curves
International Journal of Modern Science and Technology Vol. 1, No. 8, November 2016. Page 264-268. http://www.ijmst.co/ ISSN: 2456-0235. Review Article Survey Report on Space Filling Curves R. Prethee, A. R. Rishivarman Department of Mathematics, Theivanai Ammal College for Women (Autonomous) Villupuram - 605 401. Tamilnadu, India. *Corresponding author’s e-mail: [email protected] Abstract Space-filling Curves have been extensively used as a mapping from the multi-dimensional space into the one-dimensional space. Space filling curve represent one of the oldest areas of fractal geometry. Mapping the multi-dimensional space into one-dimensional domain plays an important role in every application that involves multidimensional data. We describe the notion of space filling curves and describe some of the popularly used curves. There are numerous kinds of space filling curves. The difference between such curves is in their way of mapping to the one dimensional space. Selecting the appropriate curve for any application requires knowledge of the mapping scheme provided by each space filling curve. Space filling curves are the basis for scheduling has numerous advantages like scalability in terms of the number of scheduling parameters, ease of code development and maintenance. The present paper report on various space filling curves, classifications, and its applications. It elaborates the space filling curves and their applicability in scheduling, especially in transaction. Keywords: Space filling curve, Holder Continuity, Bi-Measure-Preserving Property, Transaction Scheduling. Introduction these other curves, sometimes space-filling In mathematical analysis, a space-filling curves are still referred to as Peano curves. curve is a curve whose range contains the entire Mathematical tools 2-dimensional unit square or more generally an The Euclidean Vector Norm n-dimensional unit hypercube. -
Fractal-Based Magnetic Resonance Imaging Coils for 3T Xenon Imaging Fractal-Based Magnetic Resonance Imaging Coils for 3T Xenon Imaging
Fractal-based magnetic resonance imaging coils for 3T Xenon imaging Fractal-based magnetic resonance imaging coils for 3T Xenon imaging By Jimmy Nguyen A Thesis Submitted to the School of Graduate Studies in thePartial Fulfillment of the Requirements for the Degree Master of Applied Science McMaster University © Copyright by Jimmy Nguyen 10 July 2020 McMaster University Master of Applied Science (2020) Hamilton, Ontario (Department of Electrical & Computer Engineering) TITLE: Fractal-based magnetic resonance imaging coils for 3T Xenon imaging AUTHOR: Jimmy Nguyen, B.Eng., (McMaster University) SUPERVISOR: Dr. Michael D. Noseworthy NUMBER OF PAGES: ix, 77 ii Abstract Traditional 1H lung imaging using MRI faces numerous challenges and difficulties due to low proton density and air-tissue susceptibility artifacts. New imaging techniques using inhaled xenon gas can overcome these challenges at the cost of lower signal to noise ratio. The signal to noise ratio determines reconstructed image quality andis an essential parameter in ensuring reliable results in MR imaging. The traditional RF surface coils used in MR imaging exhibit an inhomogeneous field, leading to reduced image quality. For the last few decades, fractal-shaped antennas have been used to optimize the performance of antennas for radiofrequency systems. Although widely used in radiofrequency identification systems, mobile phones, and other applications, fractal designs have yet to be fully researched in the MRI application space. The use of fractal geometries for RF coils may prove to be fruitful and thus prompts an investiga- tion as the main goal of this thesis. Preliminary simulation results and experimental validation results show that RF coils created using the Gosper and pentaflake offer improved signal to noise ratio and exhibit a more homogeneous field than that ofa traditional circular surface coil. -
Harmonious Hilbert Curves and Other Extradimensional Space-Filling Curves
Harmonious Hilbert curves and other extradimensional space-filling curves∗ Herman Haverkorty November 2, 2012 Abstract This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d0 < d, the d-dimensional curve is compatible with the d0-dimensional curve with respect to the order in which the curves visit the points of any d0-dimensional axis-parallel space that contains the origin. Similar generalizations to arbitrary dimensions are described for several variants of Peano's curve (the original Peano curve, the coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional harmonious Hilbert curves and the Meurthe curves have neutral orientation: as compared to the curve as a whole, arbitrary pieces of the curve have each of d! possible rotations with equal probability. Thus one could say these curves are `statistically invariant' under rotation|unlike the Peano curves, the coil curves, the half-coil curves, and the familiar generalization of Hilbert curves by Butz and Moore. In addition, prompted by an application in the construction of R-trees, this paper shows how to construct a 2d-dimensional generalized Hilbert or Peano curve that traverses the points of a certain d-dimensional diagonally placed subspace in the order of a given d-dimensional generalized Hilbert or Peano curve. Pseudocode is provided for comparison operators based on the curves presented in this paper. 1 Introduction Space-filling curves A space-filling curve in d dimensions is a continuous, surjective mapping from R to Rd. -
Mathematics As “Gate-Keeper” (?): Three Theoretical Perspectives That Aim Toward Empowering All Children with a Key to the Gate DAVID W
____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 14 Number 1 Spring 2004 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA Editorial Staff A Note from the Editor Editor Dear TME readers, Holly Garrett Anthony Along with the editorial team, I present the first of two issues to be produced during my brief tenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of both Associate Editors veteran and budding scholars in mathematics education. The articles range in topic and thus invite all Ginger Rhodes those vested in mathematics education to read on. Margaret Sloan Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in Erik Tillema mathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoretical perspectives he believes can change this, and motivates mathematics educators at all levels to rethink Publication their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social Stephen Bismarck Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across Laurel Bleich chapters and applauds the picture of social justice painted by the diversity of voices therein. Dennis Hembree Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematics Advisors educators to step outside themselves and reexamine features of PhD programs and elementary Denise S. Mewborn textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom Nicholas Oppong teaching, how we engage in and write about research on or with teachers, and what features of a PhD program can inform teacher education. -
Spatial Accessibility to Amenities in Fractal and Non Fractal Urban Patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser
Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser To cite this version: Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser. Spatial accessibility to amenities in fractal and non fractal urban patterns. Environment and Planning B: Planning and Design, SAGE Publications, 2012, 39 (5), pp.801-819. 10.1068/b37132. hal-00804263 HAL Id: hal-00804263 https://hal.archives-ouvertes.fr/hal-00804263 Submitted on 14 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. TANNIER C., VUIDEL G., HOUOT H., FRANKHAUSER P. (2012), Spatial accessibility to amenities in fractal and non fractal urban patterns, Environment and Planning B: Planning and Design, vol. 39, n°5, pp. 801-819. EPB 137-132: Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile TANNIER* ([email protected]) - corresponding author Gilles VUIDEL* ([email protected]) Hélène HOUOT* ([email protected]) Pierre FRANKHAUSER* ([email protected]) * ThéMA, CNRS - University of Franche-Comté 32 rue Mégevand F-25 030 Besançon Cedex, France Tel: +33 381 66 54 81 Fax: +33 381 66 53 55 1 Spatial accessibility to amenities in fractal and non fractal urban patterns Abstract One of the challenges of urban planning and design is to come up with an optimal urban form that meets all of the environmental, social and economic expectations of sustainable urban development. -
Redalyc.Self-Similarity of Space Filling Curves
Ingeniería y Competitividad ISSN: 0123-3033 [email protected] Universidad del Valle Colombia Cardona, Luis F.; Múnera, Luis E. Self-Similarity of Space Filling Curves Ingeniería y Competitividad, vol. 18, núm. 2, 2016, pp. 113-124 Universidad del Valle Cali, Colombia Available in: http://www.redalyc.org/articulo.oa?id=291346311010 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Ingeniería y Competitividad, Volumen 18, No. 2, p. 113 - 124 (2016) COMPUTATIONAL SCIENCE AND ENGINEERING Self-Similarity of Space Filling Curves INGENIERÍA DE SISTEMAS Y COMPUTACIÓN Auto-similaridad de las Space Filling Curves Luis F. Cardona*, Luis E. Múnera** *Industrial Engineering, University of Louisville. KY, USA. ** ICT Department, School of Engineering, Department of Information and Telecommunication Technologies, Faculty of Engineering, Universidad Icesi. Cali, Colombia. [email protected]*, [email protected]** (Recibido: Noviembre 04 de 2015 – Aceptado: Abril 05 de 2016) Abstract We define exact self-similarity of Space Filling Curves on the plane. For that purpose, we adapt the general definition of exact self-similarity on sets, a typical property of fractals, to the specific characteristics of discrete approximations of Space Filling Curves. We also develop an algorithm to test exact self- similarity of discrete approximations of Space Filling Curves on the plane. In addition, we use our algorithm to determine exact self-similarity of discrete approximations of four of the most representative Space Filling Curves. -
IJR-1, Mathematics for All ... Syed Samsul Alam
January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. -
Fractal Initialization for High-Quality Mapping with Self-Organizing Maps
Neural Comput & Applic DOI 10.1007/s00521-010-0413-5 ORIGINAL ARTICLE Fractal initialization for high-quality mapping with self-organizing maps Iren Valova • Derek Beaton • Alexandre Buer • Daniel MacLean Received: 15 July 2008 / Accepted: 4 June 2010 Ó Springer-Verlag London Limited 2010 Abstract Initialization of self-organizing maps is typi- 1.1 Biological foundations cally based on random vectors within the given input space. The implicit problem with random initialization is Progress in neurophysiology and the understanding of brain the overlap (entanglement) of connections between neu- mechanisms prompted an argument by Changeux [5], that rons. In this paper, we present a new method of initiali- man and his thought process can be reduced to the physics zation based on a set of self-similar curves known as and chemistry of the brain. One logical consequence is that Hilbert curves. Hilbert curves can be scaled in network size a replication of the functions of neurons in silicon would for the number of neurons based on a simple recursive allow for a replication of man’s intelligence. Artificial (fractal) technique, implicit in the properties of Hilbert neural networks (ANN) form a class of computation sys- curves. We have shown that when using Hilbert curve tems that were inspired by early simplified model of vector (HCV) initialization in both classical SOM algo- neurons. rithm and in a parallel-growing algorithm (ParaSOM), Neurons are the basic biological cells that make up the the neural network reaches better coverage and faster brain. They form highly interconnected communication organization. networks that are the seat of thought, memory, con- sciousness, and learning [4, 6, 15]. -
FRACTAL CURVES 1. Introduction “Hike Into a Forest and You Are Surrounded by Fractals. the In- Exhaustible Detail of the Livin
FRACTAL CURVES CHELLE RITZENTHALER Abstract. Fractal curves are employed in many different disci- plines to describe anything from the growth of a tree to measuring the length of a coastline. We define a fractal curve, and as a con- sequence a rectifiable curve. We explore two well known fractals: the Koch Snowflake and the space-filling Peano Curve. Addition- ally we describe a modified version of the Snowflake that is not a fractal itself. 1. Introduction \Hike into a forest and you are surrounded by fractals. The in- exhaustible detail of the living world (with its worlds within worlds) provides inspiration for photographers, painters, and seekers of spiri- tual solace; the rugged whorls of bark, the recurring branching of trees, the erratic path of a rabbit bursting from the underfoot into the brush, and the fractal pattern in the cacophonous call of peepers on a spring night." Figure 1. The Koch Snowflake, a fractal curve, taken to the 3rd iteration. 1 2 CHELLE RITZENTHALER In his book \Fractals," John Briggs gives a wonderful introduction to fractals as they are found in nature. Figure 1 shows the first three iterations of the Koch Snowflake. When the number of iterations ap- proaches infinity this figure becomes a fractal curve. It is named for its creator Helge von Koch (1904) and the interior is also known as the Koch Island. This is just one of thousands of fractal curves studied by mathematicians today. This project explores curves in the context of the definition of a fractal. In Section 3 we define what is meant when a curve is fractal. -
Efficient Neighbor-Finding on Space-Filling Curves
Universitat¨ Stuttgart Efficient Neighbor-Finding on Space-Filling Curves Bachelor Thesis Author: David Holzm¨uller* Degree: B. Sc. Mathematik Examiner: Prof. Dr. Dominik G¨oddeke, IANS Supervisor: Prof. Dr. Miriam Mehl, IPVS October 18, 2017 arXiv:1710.06384v3 [cs.CG] 2 Nov 2019 *E-Mail: [email protected], where the ¨uin the last name has to be replaced by ue. Abstract Space-filling curves (SFC, also known as FASS-curves) are a useful tool in scientific computing and other areas of computer science to sequentialize multidimensional grids in a cache-efficient and parallelization-friendly way for storage in an array. Many algorithms, for example grid-based numerical PDE solvers, have to access all neighbor cells of each grid cell during a grid traversal. While the array indices of neighbors can be stored in a cell, they still have to be computed for initialization or when the grid is adaptively refined. A fast neighbor- finding algorithm can thus significantly improve the runtime of computations on multidimensional grids. In this thesis, we show how neighbors on many regular grids ordered by space-filling curves can be found in an average-case time complexity of (1). In 풪 general, this assumes that the local orientation (i.e. a variable of a describing grammar) of the SFC inside the grid cell is known in advance, which can be efficiently realized during traversals. Supported SFCs include Hilbert, Peano and Sierpinski curves in arbitrary dimensions. We assume that integer arithmetic operations can be performed in (1), i.e. independent of the size of the integer. -
New Thinking About Math Infinity by Alister “Mike Smith” Wilson
New thinking about math infinity by Alister “Mike Smith” Wilson (My understanding about some historical ideas in math infinity and my contributions to the subject) For basic hyperoperation awareness, try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well. However, this paper is mostly philosophical. So far as I am aware I am the first to define Nopt structures. Maybe there are several reasons for this: (1) Recursive structures can be defined by computer programs, functional powers and related fast-growing hierarchies, recurrence relations and transfinite ordinal numbers. (2) There has up to now, been no call for a geometric representation of numbers related to the Ackermann numbers. The idea of Minimal Symbolic Notation and using MSN as a sequential abstract data type, each term derived from previous terms is a new idea. Summarising my work, I can outline some of the new ideas: (1) Mixed hyperoperation numbers form interesting pattern numbers. (2) I described a new method (butdj) for coloring Catalan number trees the butdj coloring method has standard tree-representation and an original block-diagram visualisation method. (3) I gave two, original, complicated formulae for the first couple of non-trivial terms of the well-known standard FGH (fast-growing hierarchy). (4) I gave a new method (CSD) for representing these kinds of complicated formulae and clarified some technical difficulties with the standard FGH with the help of CSD notation. (5) I discovered and described a “substitution paradox” that occurs in natural examples from the FGH, and an appropriate resolution to the paradox. -
Herramientas Para Construir Mundos Vida Artificial I
HERRAMIENTAS PARA CONSTRUIR MUNDOS VIDA ARTIFICIAL I Á E G B s un libro de texto sobre temas que explico habitualmente en las asignaturas Vida Artificial y Computación Evolutiva, de la carrera Ingeniería de Iistemas; compilado de una manera personal, pues lo Eoriento a explicar herramientas conocidas de matemáticas y computación que sirven para crear complejidad, y añado experiencias propias y de mis estudiantes. Las herramientas que se explican en el libro son: Realimentación: al conectar las salidas de un sistema para que afecten a sus propias entradas se producen bucles de realimentación que cambian por completo el comportamiento del sistema. Fractales: son objetos matemáticos de muy alta complejidad aparente, pero cuyo algoritmo subyacente es muy simple. Caos: sistemas dinámicos cuyo algoritmo es determinista y perfectamen- te conocido pero que, a pesar de ello, su comportamiento futuro no se puede predecir. Leyes de potencias: sistemas que producen eventos con una distribución de probabilidad de cola gruesa, donde típicamente un 20% de los eventos contribuyen en un 80% al fenómeno bajo estudio. Estos cuatro conceptos (realimentaciones, fractales, caos y leyes de po- tencia) están fuertemente asociados entre sí, y son los generadores básicos de complejidad. Algoritmos evolutivos: si un sistema alcanza la complejidad suficiente (usando las herramientas anteriores) para ser capaz de sacar copias de sí mismo, entonces es inevitable que también aparezca la evolución. Teoría de juegos: solo se da una introducción suficiente para entender que la cooperación entre individuos puede emerger incluso cuando las inte- racciones entre ellos se dan en términos competitivos. Autómatas celulares: cuando hay una población de individuos similares que cooperan entre sí comunicándose localmente, en- tonces emergen fenómenos a nivel social, que son mucho más complejos todavía, como la capacidad de cómputo universal y la capacidad de autocopia.