Tiling-Harmonic Functions
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1 Tiling-Harmonic Functions Yilun Du Vyron Vellis (Graduate Student Mentor) Sergiy Merenkov (Faculty Mentor) PRIMES-IGL Fourth Annual PRIMES Conference May 17, 2014
2 Table of Contents 1 Grid Harmonic Function 2 Graph Harmonic Function 3 Results 4 Conclusions 5 Future 6 Acknowledgments
3 Introduction to Square Tilings Definition (Square Tilings) A square tiling T of a region D in C is a finite collection of squares with edges parallel to the x and y axes, that have mutually disjoint interiors and their union is all of D.
4 Definition of Energy Suppose that u is a function defined on the vertices of the tiling T and t is a square in T. Definition (The Oscillation of u on t) osc u (t) = max p through all vertices p on the tile t. u(p) min u(p) p
5 Definition of Energy Suppose that u is a function defined on the vertices of the tiling T and t is a square in T. Definition (The Oscillation of u on t) osc u (t) = max p through all vertices p on the tile t. Definition (The Energy of u on T ) E T (u) = t T u(p) min u(p) p osc u (t) 2.
6 A Standard Tiling Example
7 A Standard Tiling Example E = (2 0) 2 + (3 0) 2 + (3 0) 2 + (3 0) 2 = 31
8 Grid Harmonic Function Definition (Grid(Tiling) Harmonic Function) Suppose that u is defined on the boundary vertices of a tiling T. An extension of u is called harmonic if it minimizes the total energy. If we are given a infinite tiling T, such as a tiling of the upper real plane, then u is T -harmonic, if for any subtiling, u is harmonic on it.
9 Grid Harmonic Function Definition (Grid(Tiling) Harmonic Function) Suppose that u is defined on the boundary vertices of a tiling T. An extension of u is called harmonic if it minimizes the total energy. If we are given a infinite tiling T, such as a tiling of the upper real plane, then u is T -harmonic, if for any subtiling, u is harmonic on it. Theorem The function f (z) = cy is T -harmonic for every tiling T.
10 Grid Harmonic Function Definition (Grid(Tiling) Harmonic Function) Suppose that u is defined on the boundary vertices of a tiling T. An extension of u is called harmonic if it minimizes the total energy. If we are given a infinite tiling T, such as a tiling of the upper real plane, then u is T -harmonic, if for any subtiling, u is harmonic on it. Theorem The function f (z) = cy is T -harmonic for every tiling T. Remark: Energy minimizing functions are not unique.
11 An Standard Grid Tiling Energy Minimizing Example
12 An Standard Grid Tiling Energy Minimizing Example E = (2 0) 2 + (3 0) 2 + (3 1) 2 + (3 1) 2 = 21
13 Motivation for Tiling Harmonic Functions Tiling harmonic functions are analogs of carpet harmonic functions, harmonic functions defined on Sierpinski carpets.
14 Motivation for Tiling Harmonic Functions Tiling harmonic functions are analogs of carpet harmonic functions, harmonic functions defined on Sierpinski carpets. If we can prove only non-negative T -harmonic functions u that vanish on the real vertices have the form u(z) = cy, where c 0 is a constant, we might be able to generalize to carpet harmonic functions.
15 Motivation for Tiling Harmonic Functions Tiling harmonic functions are analogs of carpet harmonic functions, harmonic functions defined on Sierpinski carpets. If we can prove only non-negative T -harmonic functions u that vanish on the real vertices have the form u(z) = cy, where c 0 is a constant, we might be able to generalize to carpet harmonic functions. Through this analog on carpet harmonic functions, we might be able to generate an alternative proof of the quasisymmetric rigidity of square Sierpinski carpets.
16 Grid Harmonic Algorithm Step 1: We consider a 2 2 standard tiling, with tiles S 1,..., S 4 where we want to minimize the inner point.
17 Grid Harmonic Algorithm Step 1: We consider a 2 2 standard tiling, with tiles S 1,..., S 4 where we want to minimize the inner point. Step 2: Let M i and m i be the maximum and minimum values of each of the S i. We have the energy as E T (u) = (M 1 m 1 ) 2 + (M 2 m 2 ) 2 + (M 3 m 3 ) 2 + (M 4 m 4 ) 2 We sort all the M i and m i to be in increasing order. For example, we may have m 1 < m 2 < M 1 < m 4 < M 2 < m 3 < M 3 < M 4, We minimize the function in each interval.
18 Grid Harmonic Algorithm Step 3: Suppose that we want to find the minimum energy of the function in an interval [a, b] If b m i then let E i (x) = (M i x) 2, α i = 1 and c i = M i. If m i a b M i then let E i (x) = (M i m i ) 2, α i = 0 and c i = 0. If M i a then let E i (x) = (x m i ) 2, α i = 1 and c i = m i.
19 Grid Harmonic Algorithm Step 3: Suppose that we want to find the minimum energy of the function in an interval [a, b] If b m i then let E i (x) = (M i x) 2, α i = 1 and c i = M i. If m i a b M i then let E i (x) = (M i m i ) 2, α i = 0 and c i = 0. If M i a then let E i (x) = (x m i ) 2, α i = 1 and c i = m i. Step 4: If α α 4 = 0 then let X i = a. If α α 4 0, then define c = c 1 + c 2 + c 3 + c 4 α 1 + α 2 + α 3 + α 4, and E(x) = E 1 (x) + E 2 (x) + E 3 (x) + E 4 (x). If c is not between a and b then X k is the one of a, b that minimizes E, or a if both have the same energy. If c is between a and b then X k equals c.
20 Grid Harmonic Algorithm Algorithm for General Tiling Grid
21 Grid Harmonic Algorithm Algorithm for General Tiling Grid Step 1: We process every interior point in our general grid and find the adjacent squares of the point. Assign values to all the interior points. Randomly generate some order of interior points to loop through.
22 Grid Harmonic Algorithm Algorithm for General Tiling Grid Step 1: We process every interior point in our general grid and find the adjacent squares of the point. Assign values to all the interior points. Randomly generate some order of interior points to loop through. Step 2: Process all interior points in our grid find the minimum value of each interior point with respect to surroundings.
23 Grid Harmonic Algorithm Algorithm for General Tiling Grid Step 1: We process every interior point in our general grid and find the adjacent squares of the point. Assign values to all the interior points. Randomly generate some order of interior points to loop through. Step 2: Process all interior points in our grid find the minimum value of each interior point with respect to surroundings. Step 3: Run the algorithm again until the energy does not change significantly.
24 Grid Harmonic Algorithm Problems
25 Grid Harmonic Algorithm Problems
26 Grid Harmonic Algorithm Possible Solutions We could try perturbing the grid.
27 Grid Harmonic Algorithm Possible Solutions We could try perturbing the grid. We could also randomly generate different interior points.
28 Grid Harmonic Algorithm Possible Solutions We could try perturbing the grid. We could also randomly generate different interior points. We could also maybe set the original interior points as that of the graph harmonic function defined by the grid.
29 Grid Harmonic Algorithm Possible Solutions We could try perturbing the grid. We could also randomly generate different interior points. We could also maybe set the original interior points as that of the graph harmonic function defined by the grid. Theorem There are only a finite number of local minima of energy.
30 Graph Harmonic Function Definition (Graph Harmonic Function) A function u F(T ) is called graph-harmonic if for every interior vertex p of T, the value u(p) is equal to the average of u on the neighbor vertices of p.
31 Graph Harmonic Function Theorem A function u F(T ) is also called graph-harmonic if the sum of the squares of neighboring vertices is minimized.
32 Graph Harmonic Algorithm Algorithm for Graph Harmonic Function Step 1: We set each interior point to be the value of the average of all boundary points.
33 Graph Harmonic Algorithm Algorithm for Graph Harmonic Function Step 1: We set each interior point to be the value of the average of all boundary points. Step 2: We then loop through all interior points and set each one equal to the average of its neighbors.
34 Graph Harmonic Algorithm Algorithm for Graph Harmonic Function Step 1: We set each interior point to be the value of the average of all boundary points. Step 2: We then loop through all interior points and set each one equal to the average of its neighbors. Step 3: We repeat until values do not change significantly.
35 Visual Representation Example 1: u(0, j) = j, u(20, j) = j, u(0, i) = 0, u(20, i) = 20.
36 Visual Representation Example 2: u(0, j) = (10 j) 2, u(20, j) = (10 j) 2, u(0, i) = (10 i) 2, u(20, i) = (10 i) 2.
37 Visual Representation Example 3: u(0, j) = j, u(20, j) = 20 j, u(0, i) = i, u(20, i) = 20 i.
38 Conclusions Grid harmonic and tiling harmonic functions are not in general the same given certain boundary conditions.
39 Conclusions Grid harmonic and tiling harmonic functions are not in general the same given certain boundary conditions. Given that all 2 2 grids have their interiors minimized, it does not follow that the grid containing the tiles has minimum energy.
40 Conclusions Grid harmonic and tiling harmonic functions are not in general the same given certain boundary conditions. Given that all 2 2 grids have their interiors minimized, it does not follow that the grid containing the tiles has minimum energy. Given plane boundary conditions both tiling and grid harmonic lie on the plane.
41 Future Direction We would like to explore, using our algorithms, the general structure of grid harmonic functions.
42 Future Direction We would like to explore, using our algorithms, the general structure of grid harmonic functions. Is it true that all planes are grid harmonic?
43 Future Direction We would like to explore, using our algorithms, the general structure of grid harmonic functions. Is it true that all planes are grid harmonic? Is it true that the only bounded T -harmonic functions are constant?
44 Future Direction We would like to explore, using our algorithms, the general structure of grid harmonic functions. Is it true that all planes are grid harmonic? Is it true that the only bounded T -harmonic functions are constant? If T is a tiling of the upper complex plane and u vanishes on the real vertices and is T -harmonic, is it true that u(z) = cy?
45 Acknowledgments PRIMES Program Illinois Geometry Lab Mentor: Dr. Merenkov Graduate Mentor: Vellis Vyron Undergraduate Team Members: Qing Ma, Sufei Zhang My Parents
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