Suppose that $T$ is a square tiling in the complex plane, i.e. a finite collection of mutually disjoint squares with edges parallel to $x$ and $y$ axes. If $u$ is a function defined on the vertexes of $T$ then the energy of $u$ on $T$ is \[ E_{T}(u) = \sum_{t\in T} (\max_{v\in t} u(v) - \min_{v\in t} u(v))^2 \] where sum is taken over all squares $t$ of $T$ and max, min are taken over vertexes $v$ of $t$. Given a tiling $T$ and values on the boundary vertexes of $T$, which function defined on all vertexes of $T$ minimizes the energy? In this project we have developed a computer algorithm which calculates energy minimizers. Such minimizers are called tiling-harmonic functions. Furthermore, we compare this concept of harmonicity with the well known concept of graph-harmonic functions.
Many properties of the natural numbers can be encoded as sequences of 1's and -1's. On the surface, such sequences often show no obvious pattern and indeed seem to behave much like sequences generated by true random experiments such as coin tosses. In this project we seek to obtain a deeper understanding of the behavior of such sequences via certain "random walks" in the plane formed with these sequences. These random walks provide a natural way to visualize the degree of randomness inherent in a sequence and to detect, and possibly explain, hidden patterns, but they can also open up new mysteries that defy explanation. In Fall 2013 we focused on random walks constructed with the Moebius function, a well-known number-theoretic function with values +1, -1, and 0, and a random-like behavior that is closely connected to the Prime Number Theorem and the Riemann Hypothesis. We plan to continue these investigations in Spring 2014. For the link for further details on Prof. Hildebrand's research.
Symmetries in nature have been long studied by mathematicians, and in this project we shall be looking at two particular cases of natural symmetries: snowflakes and viruses. On the one hand, we shall study symmetries of snowflakes and other types of crystals. For the latter we will also look at how some crystals (as sugar crystals) are formed with the help of members of the Physics Department at UIUC. Although many of this objects are defined in low dimensions (i.e., snowflakes in 2 dimension) we shall look at how these concepts could be extended to higher dimensions, and produce 3D models of our proposed extensions. On the other hand, we shall consider viruses and symmetries which they present. In particular, we will consider viruses with icosahedral symmetries. Only recently have many new types of symmetries in viruses been discovered, so this area presents a very interesting branch of biology in which mathematicians could be most helpful. In this part of the project, we will study recent developments in the study of symmetries in viruses and produce models of viruses which could be used, together with the snowflakes models, in elementary or middle school with students.
If 5 electrons are placed on a metal sphere, where will they end up when they stop the motion caused by their mutual repulsion? If 5 points are placed on a sphere so that the smallest distance between any two is maximal, what are the possibilities? What if the sum of all ten distances between them is maximized? Must the center of mass be the center of the sphere? The above are problems about interacting particles. Even in the case where all particles are identical there is surprisingly little that has been proved. Some statements about electrons by Maxwell himself still constitute unsolved problems, as does the (no longer used by physicists) plum pudding model of the atom by J.J. Thomson. In this project we investigate interacting particles in two dimensions using Mathematica to visualize the resulting configurations. Symmetries abound!
The goal of our project is to design the ideal model of a Lithium-ion battery with the most efficient charging time. We analyzed two different geometric properties of configurations which are tortuosity and degrees of freedom of the particles. Based on the simulation of the charging process of batteries, we are trying to optimize the score function related to the whole system. The hope is that the score function can replace the simulation from last year and find optimal configurations of particles. We also used Python to simulate a Poisson model of the battery charging for different configurations with various positions, sizes and shapes of the particles using the finite element method (FEM). Further, we will quantify the various configurations using an appropriate objective function.
If you stand at the origin and there is a tree at each integer point in the plane, then only a fraction of the trees in this infinite forest will be visible to you. Surprisingly enough, this fraction is \frac{6}{\pi^2}. Last semester, we examined the gap distribution between these visible trees and how this distribution changes when you remove a randomly picked set of trees from the either the whole forest or the primitive forest. This semester we built on this by focusing on the distribution formed by the convolution of multiple Halls distributions. Although Halls distribution has infinite variance, we showed both numerically and by using a variation of the Central Limit Theorem that this distribution also goes to the normal distribution. We have been looking at the relationship between the distribution formed by removing randomly chosen subsets of trees from the primitive forest and the convolution of multiple Halls distributions.
Let F be a deformable quasicrystal framework in 2D or 3D. Any face of F can be made rigid by bracing it with a rhombic plate. Bracing enough faces will make the framework rigid. How many faces, and in what configuration, should we brace to make F rigid? In the 2D case the solution is obtained by associating a graph to the bracing of the framework F. For this case we present the Wester Game which allows the user to explore the deformations of the framework in the plane.
To study the 3D case we generalize the ideas we used to create the Wester Game and we present a real-time interactive animation to study coherent deformations of a cubical framework. For such a framework, we define a tunnel to be a maximal succession of adjacent cubes in the same direction. We conjecture that any 3D framework can be made rigid by plating one face in every tunnel of the framework.
Our goal is to model how the brain begins to fill in missing information in visual information. Given an occluded, or incomplete, image, we use a series of 12 CORF filters to mimic the responses of simple cells in the visual cortex. Each filter is tuned to a different preferred angle, just like the neurons they are modeling. These response values are inserted into a discrete graph that limits connections between cells to specific neighbors. The connections allow two different motions: movement along the preferred angle of the neuron and a slight change of preferred angle. We diffuse the response values using these connections in order to complete contours across occlusions in the image and fill in the missing data.
Spacing statistics measure the fine distribution of spacings between elements of an increasing sequence of finite subsets of numbers. In this project we focused on two popular spacing statistics: the gap distribution and the pair correlation. We studied experimentally the spacings between Farey sequences with divisibility constraints and between lattice angles in both Euclidean plane and hyperbolic half-plane.
Our findings are consistent with theoretical results available in certain special cases. In all situations considered in this project the limiting distribution turned out not to be random.
If we draw a pattern on a plane that repeats itself in two directions, we can study the group of symmetries of this pattern (translations, rotations, reflections, glide reflections). This group of symmetries is called a 'wallpaper group', and there are exactly 17 of them.
By taking the quotient of the plane under this group of symmetries, we get a new space called an orbifold. Our project visualizes these orbifolds and how they are obtained from a tiling of the plane.
Depicted below is a tiling of the plane, and the corresponding orbifold, which looks like a sphere with three peaks on it.
In a network where nodes are potential hosts and edges represent contact between hosts, bootstrap percolation is used to study the spread of an infection throughout the graph. Each node becomes infected if enough of its neighbors are infected. Subject to the infection rule for each vertex and some initial seed of infected nodes, we ask if the infection spreads to the whole network. The infection continues to spread iteratively until no new nodes can become infected.
Given a network and set of infection thresholds for each node, the goal is to minimize the size of the initial seed needed to infect the whole network. Our results focus on studying regular lattices with constant infection thresholds for each node and with random infection thresholds for each node. We studied the terminal states on square grid and hexagon grid, as well as the minimum initial seed and how initial seed probability affects steps to terminal state. Since the question for square lattice is mostly solved, we focused more on hexagonal lattice.