Thick Carpets. Rob Kesler and Andrew Ma. August 21, Cornell Math REU Adviser: Robert Strichartz

Transcription

1 Thick Carpets Rob Kesler and Andrew Ma Cornell Math REU Adviser: Robert Strichartz August 21, / 32

2 What is a thick carpet? 2 / 32

3 What is a thick carpet? A generalized version of Sierpinski s carpet (a self-similar fractal) 3 / 32

4 What is a thick carpet? A generalized version of Sierpinski s carpet (a self-similar fractal) Constructed by taking dividing a unit square into a 3x3 grid and removing the central square and repeating this process 4 / 32

5 What is a thick carpet? Given a sequence of odd integers n 1, n 2,... construct a Carpet as folllows: 5 / 32

6 What is a thick carpet? Given a sequence of odd integers n 1, n 2,... construct a Carpet as folllows: Divide up the unit square into an n 1 n 1 grid of cells 6 / 32

7 What is a thick carpet? Given a sequence of odd integers n 1, n 2,... construct a Carpet as folllows: Divide up the unit square into an n 1 n 1 grid of cells Remove the central cell 7 / 32

8 What is a thick carpet? Given a sequence of odd integers n 1, n 2,... construct a Carpet as folllows: Divide up the unit square into an n 1 n 1 grid of cells Remove the central cell For each cell, repeat the previous steps for the using n 2, etc. 8 / 32

9 Figure: Sierpinski Carpet Figure: Carpet 9 / 32

10 What is a thick carpet? thick = i=1 1/n i < 10 / 32

11 What is a thick carpet? thick = i=1 1/n i < Under this condition the carpet is thick enough so that there exists δ δx and δ δy for functions defined over the carpet. 11 / 32

12 What is a thick carpet? thick = i=1 1/n i < Under this condition the carpet is thick enough so that there exists δ δx and δ δy for functions defined over the carpet. Thus the usual Laplacian, = δ2 functions over thick carpets. + δ2 δx 2 δy 2 is well defined for 12 / 32

13 The project: To understand what the Laplacian is for functions defined on a thick carpets. 13 / 32

14 The project: To understand what the Laplacian is for functions defined on a thick carpets. We do this by understanding the eigenvalues and eigenfunctions of the Laplacian. 14 / 32

15 The project: To understand what the Laplacian is for functions defined on a thick carpets. We do this by understanding the eigenvalues and eigenfunctions of the Laplacian. u(x) = λu(x) λ R 15 / 32

16 The project: To understand what the Laplacian is for functions defined on a thick carpets. We do this by understanding the eigenvalues and eigenfunctions of the Laplacian. u(x) = λu(x) λ R We begin by looking at approximations of the thick Carpets 16 / 32

17 The Methods Two methods of approximations 1. Outer approximation: Create an approx. carpet using a finite sequence n 1,..., n k Calculate eigenvalues of the standard Laplacian, = δ2 f + δ2 f, δx 2 δy 2 restricted to your approx. carpet 2. Graph approximation: Create a graph of nodes and edges that models an approx. carpet. Nodes = cells Edges = shared sides of neighboring cells Compute eigenvalues of the graph Laplacian 17 / 32

18 Approx. in data Computations are done using the Finite Element Method Using a combination of Matlab built-in functions and Heilman code Currently we can collect all eigenvalues in [0, 10, 000] on batch of level 3 carpet approximations in resonable time ( 1 day) 18 / 32

19 Approx. in data Eigenvalues and Eigenfunctions are found numerically. Larger eigenvalues, i.e. eigenvalues higher in the spectrum, are sensitive to the finer detailed features of the carpet and will be distant from eigenvalues of the true carpet at low level approximations 19 / 32

20 Figure: 3 Weyl Ratio 20 / 32

21 Figure: Weyl Ratio 21 / 32

22 Approx. in data Eigenvalues and Eigenfunctions are found numerically. The eigenvalues are known to be overestimates, but they can be improved via refinements 22 / 32

23 Figure: Carpet 23 / 32

24 Figure: Carpet 24 / 32

25 Data Analysis Main Goal : identify the sequence n i s making the carpet, given an eigenspectrum 25 / 32

26 Data Analysis 26 / 32

27 Data Analysis 27 / 32

28 A First Guess Idea: For a short range at the start of the eigenspectrum, carpets with n 1 = 3 will have eigenvalues that differ by a proportionality constant. Specifically, for carpets beginning with n 1 = 3, 5 λ (n) j λ (3,5) j C (λ (3,5) j λ (n) j ) Where C = λ(3,5) 1 λ (n) 1 λ (3,5) 1 λ (3,5,7) 1 28 / 32

29 A First Guess 29 / 32

30 30 / 32

31 31 / 32

32 Acknowledgments Thanks to Rob Kesler, people in fractals 2012, Bob Strichartz, Ben, Baris, Steve Heilman, and the computer administrators. 32 / 32