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### Fractal-dimension

Course: MATH 2112, Fall 2009
School: UCF
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Word Count: 627

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Fractals Some and Fractal Dimensions The Cantor set: we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely. To calculate the fractal / Hausdorff / capacity / box-counting dimension, we see how many boxes (circles) of diameter 1/r^n we need to cover the set (in this case, we will use r = 3, since it fits nicely). D =...

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Fractals Some and Fractal Dimensions The Cantor set: we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely. To calculate the fractal / Hausdorff / capacity / box-counting dimension, we see how many boxes (circles) of diameter 1/r^n we need to cover the set (in this case, we will use r = 3, since it fits nicely). D = Lim(log(Nr)/log(1/r)) = log(2) / log(3) r 1 1/3 1/3^2 1/3^3 1/3^n Nr 1 2 2^2 2^3 2^n The Koch snowflake: We start with an equilateral triangle. We duplicate the middle third of each side, forming a smaller equilateral triangle. We repeat the process. To calculate the fractal / Hausdorff / capacity / box-counting dimension, we again see how many boxes (circles) of diameter (again)1/3^n we need to cover the set. D = Lim(log(Nr)/log(1/r)) = log(4) / log(3) r 1 1/3 1/3^2 1/3^3 1/3^n Nr 3 3*4 3 * 4^2 3 * 4^3 3 * 4^n The Sierpinski carpet: We start with a square. We remove the middle square with side one third. For each of the remaining squares of side one third, remove the central square. We repeat the process. D = Lim(log(Nr)/log(1/r)) = log(8) / log(3) r 1 1/3 1/3^2 1/3^3 1/3^n Nr 1 8 8^2 8^3 8^n The Sierpinski gasket: we do a similar process with an equilateral triangle, removing a central triangle. (Note: we could also do a similar thing taking cubes out of a larger cube -- the Sierpinski sponge -- but it's hard to draw :-) D = Lim(log(Nr)/log(1/r)) = log(3) / log(2) r 1 1/2 1/2^2 1/2^3 1/2^n Nr 1 3 3^2 3^3 3^n We can also remove other shapes. D = Lim(log(Nr)/log(1/r)) = log(4) / log(4) = 1 r 1 1/4 1/4^2 1/4^3 1/4^n Nr 1 4 4^2 4^3 4^n D = Lim(log(Nr)/log(1/r)) = log(3) / log(3) = 1 r 1 1/3 1/3^2 1/3^3 1/3^n Nr 1 3 3^2 3^3 3^n Relating this to nonlinear dynamical systems Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for ease of viewing, let's suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other -- (locally) and let's follow the state space at successive times . . . Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for ease of viewing, let's suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let's follow the state space at successive times . . . Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for ease of viewing, let's suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let's follow the state space at successive times . . . Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for ease of viewing, let's suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let's follow the state space at successive times . . . Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for ease of viewing, let's suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let's follow the state space at successive times . . . Etc. Now let's take a Poincar section (space slice) through the system Etc. Now let's take a Poincar section (space slice) through the system Etc. We are building a Cantor set (actually, a slight generalization, a Cantor dust . . .)! Etc. This sort of behavior will be generic for the stretching and folding of the state/phase space for (hyperbolic) dissipative systems . . . Etc. So, we should expect to see Cantor dusts in many Poincar sections of these sorts of systems . . . Etc. This also suggests that the fractal dimension of an attractor, or of a Poincar section of the attractor, can give us the possibility of a characteristic number, to identify the attractor, or at least to distinguish between attractors, and thus between dynamical systems . . . Etc. Fin
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UCF - MATH - 2112
Some Fractals and Fractal DimensionsThe Cantor set:we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely.To calculate the fractal / Hausdorff /capacity / box-counting
UCF - MATH - 2112
UCF - MATH - 2112
UCF - MATH - 2112
An introduction to information theory and entropyTom Carterhttp:/astarte.csustan.edu/~ tom/SFI-CSSS Complex Systems Summer School Santa FeJune, 20071ContentsMeasuring complexity Some probability ideas Basics of information theory Some entropy theory
UCF - MATH - 2112
UCF - MATH - 2112
What is Interdisciplinary?Discipline (and punish? :-)Physics ChemistryBiologyMathematicsEconomicsPsychologyEtc.Or . . .Physics Chemistry Biology Social SciencesEtc. qOr . . .MathematicsReal WorldBut is this really . . .MathematicsReal Worl
UCF - MATH - 2112
Some Fractals and Fractal Dimensions The Cantor set:we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely. To calculate the fractal / Hausdorff /capacity / box-counti
UCF - MATH - 2112
A brief survey of linear algebraTom Carterhttp:/astarte.csustan.edu/~ tom/linear-algebraSanta Fe Institute Complex Systems Summer SchoolJune, 20011Our general topics: Why linear algebra Vector spaces (ex) Examples of vector spaces (ex) Subspaces (e
UCF - MATH - 2112
The Logistic Flow(Continuous)Tom Carterhttp:/astarte.csustan.edu/ tom/SFI-CSSSComplex Systems Summer SchoolJune, 20081Logistic ow . . .We all know that the discrete logistic mapPn+1 = rPn(1 Pn)exhibits interesting behavior of various sortsfor v
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Making SenseTom Carterhttp:/astarte.csustan.edu/~ tom/SFI-CSSSApril 2, 20091Making SenseIntroduction / theme / structure 3Language and meaning Language and meaning (ex) . . . . . . . . . . . . . . .6 7Theories, models and simulation Theories, mod
UCF - MATH - 2112
Nonlinear Systems(. . . and chaos) a brief introductionTom Carter Computer Science CSU Stanislaus http:/csustan.csustan.edu/~ tom/Lecture-Notes/Nonlinear-Systems/Nonlinear-Systems.pdf November 7, 20111Our general topics:What are nonlinear systems? A
UCF - MATH - 2112
PerspectiveComplex Systems Summer School June, 2006
UCF - MATH - 2112
A Little Probability. . Coding and Information Theory Fall, 2004Tom Carter http:/astarte.csustan.edu/~ tom/ October, 20041Some probability background There are two notions of the probability of an event happening. The two general notions are: 1. A fr
UCF - MATH - 2112
A brief overview of quantum computingor, Can we compute faster in a multiverse?Tom Carterhttp:/cogs.csustan.edu/~ tom/quantum. . .. June, 20011Our general topics: Hilbert space and quantum mechanics Tensor products Quantum bits (qubits) Entangled
UCF - MATH - 5485
AIMS Exercise Set # 1Peter J. Olver1. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest positive number n1 ? The second
UCF - MATH - 5485
AIMS Exercise Set # 2Peter J. Olver1. Explain why the equation e- x = x has a solution on the interval [ 0, 1 ]. Use bisection to find the root to 4 decimal places. Can you prove that there are no other roots? 2. Find 6 3 to 5 decimal places by setting
UCF - MATH - 5485
AIMS Exercise Set # 3Peter J. Olver1. Which of the following matrices are regular? If reguolar, write down its L U 1 -2 3 2 1 0 -1 factorization. (a) , (b) , (c) -2 4 -1 . 1 4 3 -2 3 -1 2 2. In each of the following problems, find the A = L U factorizat
UCF - MATH - 5485
AIMS Exercise Set # 4Peter J. Olver1. Find the explicit formula for the solution to the following linear iterative system: u(k+1) = u(k) - 2 v (k) , v (k+1) = - 2 u(k) + v (k) , u(0) = 1, v (0) = 0.2. Determine whether or not the following matrices are
UCF - MATH - 5485
AIMS Exercise Set # 5Peter J. Olver1. Use the power method to find the dominant eigenvalue and associated 4 1 0 1 -2 0 1 1 4 1 0 eigenvector of the following matrices: (a) -3 -2 0 , (b) . 0 1 4 1 -2 5 4 1 0 1 4 2. Use Newton's Method to find all points
UCF - MATH - 5485
AIMS Exercise Set # 6Peter J. Olver1. Prove that the Midpoint Method (10.58) is a second order method. 2. Consider the initial value problem du = u(1 - u), dt for the logistic differential equation. solution for t &gt; 0. (b) Use the Euler Method with step
UCF - MATH - 5485
AIMS Exercise Set # 7Peter J. Olver1. In this exercise, you are asked to find &quot;one-sided&quot; finite difference formulas for derivatives. These are useful for approximating derivatives of functions at or near the boundary of their domain. (a) Construct a se
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver6. Eigenvalues and Singular ValuesIn this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint, the eigenvectors indicate the directions of pure stretch and t
UCF - MATH - 5485
Chapter 16 Complex AnalysisThe term &quot;complex analysis&quot; refers to the calculus of complex-valued functions f (z) depending on a single complex variable z. On the surface, it may seem that this subject should merely be a simple reworking of standard real v
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver1. Computer ArithmeticThe purpose of computing is insight, not numbers. - R.W. Hamming, [23]The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathemati
UCF - MATH - 5485
Chapter 21 The Calculus of VariationsWe have already had ample opportunity to exploit Nature's propensity to minimize. Minimization principles form one of the most wide-ranging means of formulating mathematical models governing the equilibrium configurat
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver3. Review of Matrix AlgebraVectors and matrices are essential for modern analysis of systems of equations - algebrai, differential, functional, etc. In this part, we will review the most basic facts of matrix arithm
UCF - MATH - 5485
Chapter 13 Fourier AnalysisIn addition to their inestimable importance in mathematics and its applications, Fourier series also serve as the entry point into the wonderful world of Fourier analysis and its wide-ranging extensions and generalizations. An
UCF - MATH - 5485
Chapter 12 Fourier SeriesJust before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his investigations into the partial differential equations modeling vibration and heat propa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver4. Gaussian EliminationIn this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor of one of the all-time mathematical greats - the early nin
UCF - MATH - 5485
Chapter 14 Vibration and Diffusion in OneDimensional MediaIn this chapter, we study the solutions, both analytical and numerical, to the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffusion of thermal
UCF - MATH - 5485
Math 5485 September 11, 2006Homework #1Problems: 1.3 1.4 2.1 1b, 3, 4a,b(single precision only), 9. 1a, 2, 7, 13. 1d, 3, 8, 11, 16a.Due: Wednesday, September 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 September 20, 2006Homework #2Problems: 2.2 2.3 2.4 1d, 5 (only do parts 1 &amp; 3), 13. 1, 5, 7, 11. 1d, 4, 9, 14a.Due: Wednesday, September 27 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 September 27, 2006Homework #3Problems: 2.5 2.6 1d, 6, 11a. 1, 5, 8.Due: Wednesday, October 4 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 October 4, 2006Homework #4Problems: 3.1 3.2 3.5 1, 8, 10, 12b. 7a, 14, 18b. 10a, 11.Due: Friday, October 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm: Wednesday, November 1 Will cover chapters 1, 2, 3. You
UCF - MATH - 5485
Math 5485 October 13, 2006Homework #5Problems: 3.3 3.6 3.7 2a, 3a, b(a), c, 5b, d (also, what is the spectral radius?), 6b, c, 7a, 10. 2,10. 14b, 19.Due: Friday, October 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm:
UCF - MATH - 5485
Math 5485 October 23, 2006Homework #6Problems: 3.7 3.8 5b, 6. 3a, 9, 11 (for 9), 12 (for 9), 13.Due: Monday, November 6 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm: Wednesday, November 1 Will cover sections 1.24, 2.16,
UCF - MATH - 5485
Math 5485 November 15, 2006Homework #7Problems: 3.10 4.1 4.2 4.3 7 (just do Newton's Method), 11b. 2, 11, 14a, 15a. 2, 8, 10. 1b, 5.Due: Monday, November 27 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 November 27, 2006Homework #8Problems: 4.4 4.5 1ac, 4a, 5b, 8. 6 (ignore the Wilkinson shift), 12 (compare the convergence rate of the direct QR algorithm with that based on tridiagonalization).Due: Monday, December 4 Text: B. Bradie, A Friend
UCF - MATH - 5485
Math 5485 December 4, 2006Homework #9Problems: 5.1 5.2 5.3 5.4 2, 5, 8. 1b, 4, 9, 12. 4, 8, 11. 2, 3, 10 (only uniform and Chebyshev).Due: Wednesday, December 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis. Second Midterm: Friday, De
UCF - MATH - 5485
Chapter 17 Dynamics of Planar MediaIn this chapter, we continue our ascent of the dimensional ladder for linear systems. In Chapter 6, we embarked on our journey with equilibrium configurations of discrete systems - massspring chains, circuits, and struc
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver7. Iterative Methods for Linear SystemsLinear iteration coincides with multiplication by successive powers of a matrix; convergence of the iterates depends on the magnitude of its eigenvalues. We discuss in some det
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver5. Inner Products and NormsThe norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number of other important norms that are used in numerical analysis
UCF - MATH - 5485
Chapter 15 The Planar Laplace EquationThe fundamental partial differential equations that govern the equilibrium mechanics of multi-dimensional media are the Laplace equation and its inhomogeneous counterpart, the Poisson equation. The Laplace equation i
UCF - MATH - 5485
Very Basic MATLABPeter J. Olver January, 2009 Matrices: Type your matrix as follows: Use space or , to separate entries, and ; or return after each row. &gt; A = [4 5 6 -9;5 0 -3 6;7 8 5 0; -1 4 5 1] or &gt; A = [4,5,6,-9;5,0,-3,6;7,8,5,0;-1,4,5,1] or &gt; A = [
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver9. Numerical Solution of Algebraic SystemsIn this part, we discuss basic iterative methods for solving systems of algebraic equations. By far the most common is a vector-valued version of Newton's Method, which will
UCF - MATH - 5485
Chapter 19 Nonlinear SystemsNonlinearity is ubiquitous in physical phenomena. Fluid and plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inhe
UCF - MATH - 5485
Chapter 22 Nonlinear Partial Differential EquationsThe ultimate topic to be touched on in this book is the vast and active field of nonlinear partial differential equations. Leaving aside quantum mechanics, which remains to date an inherently linear theo
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver11. Numerical Solution of the Heat and Wave EquationsIn this part, we study numerical solution methodss for the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffus
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver10. Numerical Solution of Ordinary Differential EquationsThis part is concerned with the numerical solution of initial value problems for systems of ordinary differential equations. We will introduce the most basic
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver8. Numerical Computation of EigenvaluesIn this part, we discuss some practical methods for computing eigenvalues and eigenvectors of matrices. Needless to say, we completely avoid trying to solve (or even write down
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver13. Approximation and InterpolationWe will now apply our minimization results to the interpolation and least squares fitting of data and functions.13.1. Least Squares.Linear systems with more equations than unknow
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver2. Numerical Solution of Scalar EquationsMost numerical solution methods are based on some form of iteration. The basic idea is that repeated application of the algorithm will produce closer and closer approximation
UCF - MATH - 5485
Chapter 20 Nonlinear Ordinary Differential EquationsThis chapter is concerned with initial value problems for systems of ordinary differential equations. We have already dealt with the linear case in Chapter 9, and so here our emphasis will be on nonline
UCF - MATH - 5485
Chapter 18 Partial Differential Equations in ThreeDimensional SpaceAt last we have ascended the dimensional ladder to its ultimate rung (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. A
UCF - MATH - 5485
Orthogonal Bases and the QR Algorithmby Peter J. Olver University of Minnesota1. Orthogonal Bases.Throughout, we work in the Euclidean vector space V = R n , the space of column vectors with n real entries. As inner product, we will only use the dot pr
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver14. Finite ElementsIn this part, we introduce the powerful finite element method for finding numerical approximations to the solutions to boundary value problems involving both ordinary and partial differential equa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver12. MinimizationIn this part, we will introduce and solve the most basic mathematical optimization problem: minimize a quadratic function depending on several variables. This will require a short introduction to pos
UCF - MATH - 5587
Remark : On a connected domain R 2 , all harmonic conjugates to a given function u(x, y) only differ by a constant: v(x, y) = v(x, y) + c; see Exercise . Although most harmonic functions have harmonic conjugates, unfortunately this is not always the case.
UCF - MATH - 5587
Chapter 7 Complex Analysis and Conformal MappingThe term &quot;complex analysis&quot; refers to the calculus of complex-valued functions f (z) depending on a single complex variable z. To the novice, it may seem that this subject should merely be a simple reworkin
UCF - MATH - 5587
1 Re z Figure 7.1.1 Im z 1 Real and Imaginary Parts of f (z) = z .Therefore, if f (z) is any complex function, we can write it as a complex combination f (z) = f (x + i y) = u(x, y) + i v(x, y), of two inter-related real harmonic functions: u(x, y) = Re
UCF - MATH - 5587
Figure 7.4.Real and Imaginary Parts ofz.also have complex logarithms! On the other hand, if z = x &lt; 0 is real and negative, then log z = log | x | + (2 k + 1) i is complex no matter which value of ph z is chosen. (This explains why one avoids defining