Critical Phenomena and Percolation Theory: I


 Ursula Henderson
 2 years ago
 Views:
Transcription
1 Critical Phenomena and Percolation Theory: I Kim Christensen Complexity & Networks Group Imperial College London Joint CRMImperial College School and Workshop Complex Systems Barcelona 813 April 2013
2 Outline Critical Phenomena & Percolation Theory 1 Critical Phenomena & Percolation Theory 2 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size 3
3 Critical Phenomena & Percolation Theory Aim: Study connections between macroscopic quantities and the underlying microscopic world in a model displaying a phase transition. Objective: Gain qualitative and quantitative understanding of critical phenomena and associated concepts such as scalefree behaviour, scaling theory, and universality.
4 L L Each site in a (regular) lattice is occupied randomly and independently with occupation probability p, 0 p 1. A cluster is a group of nearestneighbour occupied sites. The size s of a cluster is the number of sites in the cluster. The critical occupation probability p c is the occupation probability p at which an infinite cluster appears for the first time in an infinite lattice L =.
5 Percolation deals with the number of the clusters formed properties of the clusters formed when occupying randomly and independently each site in a lattice with probability p.
6 Quantities of interest Onset of percolation critical occupation probability, p c. Probability that a site belongs to the infinite cluster, P (p). Geometry of the infinite cluster at p = p c and p > p c. Excluding the infinite cluster: Average cluster size, χ(p). Typical size of the largest cluster, s ξ (p). Typical radius (linear size) of the largest cluster, ξ(p).
7 For fixed lattice size L, there is only one parameter, the occupation probability p, 0 p 1. p = 0: Empty lattice. No clusters. 0 < p < 1: Percolation is a random{ process. No. of different realisations 2 L L 10 6,773 for L=150; ,370 for L=600. p = 1: Fully occupied lattice. One cluster of size s = L 2. A percolating cluster is one that spans the lattice from left to right, top to bottom, or both.
8 p = 0.10 p = 0.55 p = 0.58 p = p c p = 0.65 p = 0.90
9 Typically expects no percolating cluster for small p. Typically expects a percolating cluster for large p. Consider these two probabilities at occupation probability p in a lattice of size L: Π (p; L) = prob. that percolating cluster exists. P (p; L) = prob. that a site belongs to a percolating cluster = fraction of volume covered by percolating cluster
10 Prob. that percolating cluster exists at occupation probability p Π (p, L = ) p
11 Prob. that percolating cluster exists at occupation probability p Π (p, L = ) p
12 Prob. that percolating cluster exists at occupation probability p Π (p, L = ) p
13 1 0.8 Π (p, L = ) No percolating cluster Subcritical p<p c Phase transition Critical p=p c Percolating cluster Supercritical p>p c p Critical point
14 Π (p; L = ) = { 0 for p < p c 1 for p > p c The critical occupation probability p c is the occupation probability above which a percolating (infinite) cluster appears for the first time in an infinite lattice. Onset of percolation is a geometrical phase transition: When increasing p from 0 towards 1, there is a phase transition at p = p c from a lattice with no percolating infinite cluster for p < p c to a lattice with a percolating infinite cluster for p > p c. For twodimensional square lattice p c = For twodimensional triangular lattice p c = 1/2. For threedimensional simple cubic lattice p c =
15 Prob. that site belongs to percolating cluster at occ. prob. p. 1 P (p; L= ) picks up abruptly for p>p c P 0.8 (p; L= ) = 0 for p p c P (p, L = ) No percolating cluster Percolating cluster p
16 P (p; L = ) = = { 0 for p p c nonzero for p > p c { 0 for p p c A (p p c ) β for p p c +. The critical exponent β characterises the abrupt pickup of the order parameter P (p) for p p + c.
17 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) 0 p 0 p c 1
18 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) 0 p 0 p c 1
19 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) 0 p 0 p c 1
20 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) s ξ (p) p p c 1/σ for p p c Critical exponent σ = 36 in d = σ is independent of lattice details Example of universality 0 p 0 p c 1
21 Excluding the percolating (infinite) cluster, what is the typical radius (linear size) of the largest cluster ξ(p)? ξ(p) ξ(p) p p c ν for p p c Critical exponent ν = 4 in d =2 3 ν is independent of lattice details Example of universality 0 p c 1 p
22 Excluding the percolating (infinite) cluster, what is the average cluster size to which an occupied site belongs, χ(p)? χ(p) χ(p) p p c γ for p p c Critical exponent γ = 43 in d = γ is independent of lattice details Example of universality 0 p 0 p c 1
23 l 10 8 l M (pc; l) l Mass of the percolating cluster at p = p c increases with window size l: M (p c ; l) l D for l 1. Critical exponent D is the fractal dimension of cluster. D = 91 in d = D is independent of lattice details. Example of universality.
24 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size Can be solved analytically. Many of the characteristic features encountered are present for percolation in d > 1. L What is the critical occupation probability p c?
25 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size Prob. that site belongs to percolating infinite cluster P (p) For d = 1, p c = p
26 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size Calculate the cluster size frequency N(s, p; L) probabilistically. Empty site s consecutive sites occupied s Empty site Cluster number density = number of sclusters per lattice site: n(s, p) = lim L N(s,p:L) L =(prob. empty site) (prob. s occupied sites) (prob. empty site) = (1 p)p s (1 p) = (1 p) 2 p s
27 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size 10 0 n(s, p) p = 0.4 p = p = 0.99 p = p = s
28 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size n(s, p) = (1 p) 2 p s with the characteristic cluster size s ξ (p)= 1 ln p = = (1 p) 2 exp (ln p s ) = (1 p) 2 exp (s ln p) = (1 p) 2 exp ( s/s ξ ), 1 ln (1 [1 p]) (1 p) 1 = (p c p) 1 for p p c. For d = 1, the critical exponent σ = 1.
29 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size sξ(p) p
30 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size sξ(p) (1 p)
31 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size The probability that a site belongs to an scluster: sn(s, p). Given an occupied site  how large, on average, is its cluster? s=1 χ(p) = s2 n(s, p) s=1 sn(s, p) = 1 + p (see page 11 in notes) 1 p 2(1 p) 1 for p 1
32 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size χ(p) p
33 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size χ(p) (1 p)
34 General pattern for the exact solution to d = 1 percolation: Characteristic cluster size s ξ (p) = 1 ln p (p c p) 1 for p p c, σ = 1. Average cluster size χ(p) = 1+p 1 p 2 (p c p) 1 for p p c, γ = 1. Asymptotically close to p c, the divergence is characterized by a powerlaw in (p c p), the distance away from the critical point. Special for d = 1 is that the phasetransition can only be approached from below, p p c.
35 When increasing occupation probability p from 0 towards 1, there is a phase transition at p = p c from a lattice with no percolating infinite cluster for p < p c to a lattice with a percolating infinite cluster for p > p c. Subcritical behaviour for p < p c where ξ <. Critical behaviour for p = p c where ξ =. Supercritical behaviour for p > p c where ξ <. Order parameter picks up abruptly at p = p c : P (p) (p p c ) β for p p + c. Quantities of interest diverges at p = p c : Characteristic cluster size: s ξ (p) p p c 1/σ for p p c. Average cluster size: χ(p) p p c γ for p p c. Typical radius of largest cluster: ξ(p) p p c ν for p p c.
36 Thank you for listening! For a comprehensive introduction to percolation, please see K. Christensen and N.R. Moloney, Complexity and Criticality, Imperial College Press (2005), Chapter 1. Access to animations, please visit
A boxcovering algorithm for fractal scaling in scalefree networks
CHAOS 17, 026116 2007 A boxcovering algorithm for fractal scaling in scalefree networks J. S. Kim CTP & FPRD, School of Physics and Astronomy, Seoul National University, NS50, Seoul 151747, Korea K.I.
More informationFIELD THEORY OF ISING PERCOLATING CLUSTERS
UK Meeting on Integrable Models and Conformal Field heory University of Kent, Canterbury 1617 April 21 FIELD HEORY OF ISING PERCOLAING CLUSERS Gesualdo Delfino SISSArieste Based on : GD, Nucl.Phys.B
More informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationGrade 5 Work Sta on Perimeter, Area, Volume
Grade 5 Work Sta on Perimeter, Area, Volume #ThankATeacher #TeacherDay #TeacherApprecia onweek 6. 12. Folder tab label: RC 3 TEKS 5(4)(H) Perimeter, Area, and Volume Cover: Reporting Category 3 Geometry
More informationSelf similarity of complex networks & hidden metric spaces
Self similarity of complex networks & hidden metric spaces M. ÁNGELES SERRANO Departament de Química Física Universitat de Barcelona TERANET: Toward Evolutive Routing Algorithms for scalefree/internetlike
More informationRheinische FriedrichWilhelmsUniversität Bonn Master Course, WS 2010/2011. Computational Physics Project
Rheinische FriedrichWilhelmsUniversität Bonn Master Course, WS 2010/2011 Computational Physics Project Title: Fractal Growth Authors: Anton Iakovlev & Martin Garbade Examiner: Prof. Carsten Urbach Tutor:
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationPrentice Hall Connected Mathematics 2, 7th Grade Units 2009
Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationNEW MEXICO Grade 6 MATHEMATICS STANDARDS
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical
More informationMeasuring Line Edge Roughness: Fluctuations in Uncertainty
Tutor6.doc: Version 5/6/08 T h e L i t h o g r a p h y E x p e r t (August 008) Measuring Line Edge Roughness: Fluctuations in Uncertainty Line edge roughness () is the deviation of a feature edge (as
More informationBig Data Analytics of MultiRelationship Online Social Network Based on MultiSubnet Composited Complex Network
, pp.273284 http://dx.doi.org/10.14257/ijdta.2015.8.5.24 Big Data Analytics of MultiRelationship Online Social Network Based on MultiSubnet Composited Complex Network Gengxin Sun 1, Sheng Bin 2 and
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationBondcorrelated percolation model and the unusual behaviour of supercooled water
J. Phys. A: Math. Gen. 16 (1983) L321L326. Printed in Great Britain LETTER TO THE EDITOR Bondcorrelated percolation model and the unusual behaviour of supercooled water ChinKun Hu LashMiller Chemical
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More informationPerformance Level Descriptors Grade 6 Mathematics
Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.12 Grade 6 Math : SubClaim A The student solves problems involving the Major Content for grade/course with
More informationStudents will understand 1. use numerical bases and the laws of exponents
Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?
More informationCommon Core State Standards  Mathematics Content Emphases by Cluster Grade K
Grade K Not all of the content in a given grade is emphasized equally in the standards. Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to
More informationIntroduction to the Mathematics Correlation
Introduction to the Mathematics Correlation Correlation between National Common Core Standards for Mathematics and the North American Association for Environmental Education Guidelines for Excellence in
More informationApproaches for Analyzing Survey Data: a Discussion
Approaches for Analyzing Survey Data: a Discussion David Binder 1, Georgia Roberts 1 Statistics Canada 1 Abstract In recent years, an increasing number of researchers have been able to access survey microdata
More informationSamples of Allowable Supplemental Aids for STAAR Assessments. Updates from 12/2011
Samples of Allowable Supplemental Aids for STAAR Assessments Updates from 12/2011 All Subjects: Mnemonic Devices A mnemonic device is a learning technique that assists with memory. Only mnemonic devices
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationDmitri Krioukov CAIDA/UCSD
Hyperbolic geometry of complex networks Dmitri Krioukov CAIDA/UCSD dima@caida.org F. Papadopoulos, M. Boguñá, A. Vahdat, and kc claffy Complex networks Technological Internet Transportation Power grid
More informationHydrodynamic Limits of Randomized Load Balancing Networks
Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli
More informationDiffusion and Conduction in Percolation Systems Theory and Applications
Diffusion and Conduction in Percolation Systems Theory and Applications Armin Bunde and Jan W. Kantelhardt 1 Introduction Percolation is a standard model for disordered systems. Its applications range
More informationChapter 29 ScaleFree Network Topologies with Clustering Similar to Online Social Networks
Chapter 29 ScaleFree Network Topologies with Clustering Similar to Online Social Networks Imre Varga Abstract In this paper I propose a novel method to model real online social networks where the growing
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationBackbone and elastic backbone of percolation clusters obtained by the new method of burning
J. Phys. A: Math. Gen. 17 (1984) L261L266. Printed in Great Britain LE ITER TO THE EDITOR Backbone and elastic backbone of percolation clusters obtained by the new method of burning H J HerrmanntS, D
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationA Coefficient of Variation for Skewed and HeavyTailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University
A Coefficient of Variation for Skewed and HeavyTailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a
More informationMathematics Interim Assessment Blocks Blueprint V
67 Blueprint V.5.7.6 The Smarter Balanced Interim Assessment Blocks (IABs) are one of two distinct types of interim assessments being made available by the Consortium; the other type is the Interim Comprehensive
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More informationA STOCHASTIC MODEL FOR THE SPREADING OF AN IDEA IN A HUMAN COMMUNITY
6th Jagna International Workshop International Journal of Modern Physics: Conference Series Vol. 7 (22) 83 93 c World Scientific Publishing Company DOI:.42/S29452797 A STOCHASTIC MODEL FOR THE SPREADING
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More informationarxiv:physics/0607202v2 [physics.compph] 9 Nov 2006
Stock price fluctuations and the mimetic behaviors of traders Junichi Maskawa Department of Management Information, Fukuyama Heisei University, Fukuyama, Hiroshima 7200001, Japan (Dated: February 2,
More informationPennsylvania System of School Assessment
Pennsylvania System of School Assessment The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationFor example, estimate the population of the United States as 3 times 10⁸ and the
CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number
More informationMath at a Glance for April
Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common
More informationThe Power (Law) of Indian Markets: Analysing NSE and BSE Trading Statistics
The Power (Law) of Indian Markets: Analysing NSE and BSE Trading Statistics Sitabhra Sinha and Raj Kumar Pan The Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai  6 113, India. sitabhra@imsc.res.in
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationAnNajah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)
AnNajah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 5 Example: Joint
More informationCommon Core State Standards for Mathematics Accelerated 7th Grade
A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting
More informationGeorgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More information21. Channel flow III (8.10 8.11)
21. Channel flow III (8.10 8.11) 1. Hydraulic jump 2. Nonuniform flow section types 3. Step calculation of water surface 4. Flow measuring in channels 5. Examples E22, E24, and E25 1. Hydraulic jump Occurs
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationImplications of an inverse branching aftershock sequence model
Implications of an inverse branching aftershock sequence model D. L. Turcotte* and S. G. Abaimov Department of Geology, University of California, Davis, California 95616, USA I. Dobson Electrical and Computer
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationModeling in Geometry
Modeling in Geometry Overview Number of instruction days: 810 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Use geometric shapes and their components to represent
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationTrading and Price Diffusion: Stock Market Modeling Using the Approach of Statistical Physics Ph.D. thesis statements. Supervisors: Dr.
Trading and Price Diffusion: Stock Market Modeling Using the Approach of Statistical Physics Ph.D. thesis statements László Gillemot Supervisors: Dr. János Kertész Dr. Doyne Farmer BUDAPEST UNIVERSITY
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationClaudio J. Tessone. Pau Amengual. Maxi San Miguel. Raúl Toral. Horacio Wio. Eur. Phys. J. B 39, 535 (2004) http://www.imedea.uib.
Horacio Wio Raúl Toral Eur. Phys. J. B 39, 535 (2004) Claudio J. Tessone Pau Amengual Maxi San Miguel http://www.imedea.uib.es/physdept Models of Consensus vs. Polarization, or Segregation: Voter model,
More informationDensity Determinations and Various Methods to Measure
Density Determinations and Various Methods to Measure Volume GOAL AND OVERVIEW This lab provides an introduction to the concept and applications of density measurements. The densities of brass and aluminum
More informationVisualization of General Defined Space Data
International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 013 Visualization of General Defined Space Data John R Rankin La Trobe University, Australia Abstract A new algorithm
More informationDave Sly, PhD, MBA, PE Iowa State University
Dave Sly, PhD, MBA, PE Iowa State University Tuggers deliver to multiple locations on one trip, where Unit Load deliveries involve only one location per trip. Tugger deliveries are more complex since the
More informationUniversal hashing. In other words, the probability of a collision for two different keys x and y given a hash function randomly chosen from H is 1/m.
Universal hashing No matter how we choose our hash function, it is always possible to devise a set of keys that will hash to the same slot, making the hash scheme perform poorly. To circumvent this, we
More informationAppendix 3 IB Diploma Programme Course Outlines
Appendix 3 IB Diploma Programme Course Outlines The following points should be addressed when preparing course outlines for each IB Diploma Programme subject to be taught. Please be sure to use IBO nomenclature
More informationA wave lab inside a coaxial cable
INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 25 (2004) 581 591 EUROPEAN JOURNAL OF PHYSICS PII: S01430807(04)76273X A wave lab inside a coaxial cable JoãoMSerra,MiguelCBrito,JMaiaAlves and A M Vallera
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationChapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
More informationDisorderinduced rounding of the phase transition. in the largeqstate Potts model. F. Iglói SZFKI  Budapest
Disorderinduced rounding of the phase transition in the largeqstate Potts model M.T. Mercaldo JC. Anglès d Auriac Università di Salerno CNRS  Grenoble F. Iglói SZFKI  Budapest Motivations 2. CRITICAL
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationFundamentals of grain boundaries and grain boundary migration
1. Fundamentals of grain boundaries and grain boundary migration 1.1. Introduction The properties of crystalline metallic materials are determined by their deviation from a perfect crystal lattice, which
More informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More informationMcDougal Littell California:
McDougal Littell California: PreAlgebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California PreAlgebra Components: Pupil Edition (PE), Teacher s Edition (TE),
More informationClass 3. 1. General Aptitude test. Qualitative Reasoning Quantitative reasoning Language Conventions. 2. Mental Mathematics
Subject Class 3 Chapters 2. Mental Mathematics Place the value, Stop the clock, Dice, Count my bills, Division, Fractions, Pipe line, Square maker, I.Q. Test, Identify shapes, Make series, Angel game,
More informationRatios and Proportional Relationships Grade 6 Grade 7 Grade 8
Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship
More informationData Preparation and Statistical Displays
Reservoir Modeling with GSLIB Data Preparation and Statistical Displays Data Cleaning / Quality Control Statistics as Parameters for Random Function Models Univariate Statistics Histograms and Probability
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationAnswer Key For The California Mathematics Standards Grade 7
Introduction: Summary of Goals GRADE SEVEN By the end of grade seven, students are adept at manipulating numbers and equations and understand the general principles at work. Students understand and use
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationCCGPS Curriculum Map. Mathematics. 7 th Grade
CCGPS Curriculum Map Mathematics 7 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Unit 1 Operations with Rational Numbers a b
More informationChapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School
Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education
More informationarxiv:condmat/9910374v1 [condmat.statmech] 23 Oct 1999
accepted for publication in Physical Review E. 1 Moment analysis of the probability distributions of different sandpile models arxiv:condmat/9910374v1 [condmat.statmech] 23 Oct 1999 S. Lübeck Theoretische
More informationThe GeoMedia Fusion Validate Geometry command provides the GUI for detecting geometric anomalies on a single feature.
The GeoMedia Fusion Validate Geometry command provides the GUI for detecting geometric anomalies on a single feature. Below is a discussion of the Standard Advanced Validate Geometry types. Empty Geometry
More informationangle Definition and illustration (if applicable): a figure formed by two rays called sides having a common endpoint called the vertex
angle a figure formed by two rays called sides having a common endpoint called the vertex area the number of square units needed to cover a surface array a set of objects or numbers arranged in rows and
More informationMachine Learning and Pattern Recognition Logistic Regression
Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationSQUARES AND SQUARE ROOTS
1. Squares and Square Roots SQUARES AND SQUARE ROOTS In this lesson, students link the geometric concepts of side length and area of a square to the algebra concepts of squares and square roots of numbers.
More informationEXAM. Practice for Third Exam. Math , Fall Dec 1, 2003 ANSWERS
EXAM Practice for Third Exam Math 35006, Fall 003 Dec, 003 ANSWERS i Problem series.) A.. In each part determine if the series is convergent or divergent. If it is convergent find the sum. (These are
More informationSupporting Material to Crowding of molecular motors determines microtubule depolymerization
Supporting Material to Crowding of molecular motors determines microtubule depolymerization Louis Reese Anna Melbinger Erwin Frey Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,
More informationStatic analysis of parity games: alternating reachability under parity
8 January 2016, DTU Denmark Static analysis of parity games: alternating reachability under parity Michael Huth, Imperial College London Nir Piterman, University of Leicester Jim HuanPu Kuo, Imperial
More informationGraph theoretic techniques in the analysis of uniquely localizable sensor networks
Graph theoretic techniques in the analysis of uniquely localizable sensor networks Bill Jackson 1 and Tibor Jordán 2 ABSTRACT In the network localization problem the goal is to determine the location of
More informationAcademic Standards for Mathematics
Academic Standards for Grades Pre K High School Pennsylvania Department of Education INTRODUCTION The Pennsylvania Core Standards in in grades PreK 5 lay a solid foundation in whole numbers, addition,
More informationVocabulary Cards and Word Walls Revised: June 29, 2011
Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,
More informationThreeDimensional Redundancy Codes for Archival Storage
ThreeDimensional Redundancy Codes for Archival Storage JehanFrançois Pâris Darrell D. E. Long Witold Litwin Department of Computer Science University of Houston Houston, T, USA jfparis@uh.edu Department
More informationNEW GENERATION OF COMPUTER AIDED DESIGN IN SPACE PLANNING METHODS A SURVEY AND A PROPOSAL
NEW GENERATION OF COMPUTER AIDED DESIGN IN SPACE PLANNING METHODS A SURVEY AND A PROPOSAL YINGCHUN HSU, ROBERT J. KRAWCZYK Illinois Institute of Technology, Chicago, IL USA Email address: hsuying1@iit.edu
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationSphere Packings, Lattices, and Kissing Configurations in R n
Sphere Packings, Lattices, and Kissing Configurations in R n Stephanie Vance University of Washington April 9, 2009 Stephanie Vance (University of Washington)Sphere Packings, Lattices, and Kissing Configurations
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationa. Look under the menu item Introduction to see how the standards are organized by Standards, Clusters and Domains.
Chapter One Section 1.1 1. Go to the Common Core State Standards website (http://www.corestandards.org/math). This is the main site for further questions about the Common Core Standards for Mathematics.
More informationClassification Problems
Classification Read Chapter 4 in the text by Bishop, except omit Sections 4.1.6, 4.1.7, 4.2.4, 4.3.3, 4.3.5, 4.3.6, 4.4, and 4.5. Also, review sections 1.5.1, 1.5.2, 1.5.3, and 1.5.4. Classification Problems
More informationEdmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).
INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes
More informationMeaneld theory for scalefree random networks
Physica A 272 (1999) 173 187 www.elsevier.com/locate/physa Meaneld theory for scalefree random networks AlbertLaszlo Barabasi,Reka Albert, Hawoong Jeong Department of Physics, University of NotreDame,
More informationGRADE 7 SKILL VOCABULARY MATHEMATICAL PRACTICES Add linear expressions with rational coefficients. 7.EE.1
Common Core Math Curriculum Grade 7 ESSENTIAL QUESTIONS DOMAINS AND CLUSTERS Expressions & Equations What are the 7.EE properties of Use properties of operations? operations to generate equivalent expressions.
More information