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Math Project Descriptions 2013 (original description)

Algebraic Graph Theory Group  Dr. Sung-Yell Song, Katy Nowak
 Existence and Nonexistence of (Directed) Strongly Regular Graphs

A strongly-regular graph with parameters (v, k, λ, μ) is defined as an undirected simple graph G with v vertices satisfying the property:
“The number of common neighbors of vertices x and y is k if x = y, λ if x and y are adjacent, and μ if x and y are non-adjacent vertices.”
Let A denote the adjacency matrix of G, and let I and J  denote the v x v identity matrix and all-ones matrix, respectively. Then G is a strongly-regular graph with parameters (v, k, λ, μ) if and only if
(i) JA = AJ = kJ and
(ii) A^2 = kI + λA + μ(J-I-A)

A loopless directed graph D with v vertices and adjacency matrix A is called directed strongly-regular graph with parameters (v, k, t, λ, μ)  if and only if A satisfies the following conditions:
(i) JA = AJ = kJ and
(ii) A^2 = tI + λA + μ(J-I-A)

We are interested in constructing (directed) strongly-regular Cayley graphs of various classical groups with suitable generator sets. We are also interested in settling some problems related to characterization and classification of these graphs and related objects. So, the sample problems that we are tempted to
explore look like:
• Find (directed) strongly-regular graphs that can be obtained as Cayley graphs of classical groups.
• How many strongly regular graphs with parameters (64, 28, 12, 12) are there?
• Is there a directed strongly regular graph with parameters (24, 10, 5, 3, 5)?
• Characterize all tactical configurations that yield the directed strongly regular graphs with parameters ((n² − 1)(n³ − 1)/(n − 1)², n(n + 1), n, n − 1, n).


Analysis Group    Dr. Justin R. Peters, Jiali Li
We will be investigating two unrelated problems. Both problems will require some background in introductory analysis. One of the problems will have a large computational compenent. Students may choose to be involved in one or both projects.

1) Given a transcendental function f(x), let p_n(x) be its nth Taylor polynomial. We will investigate the behavoir of the zeroes of pn in the complex plane. We will be asking questions such as, do the zeroes of p_n lie in some region we can determine? Do the zeroes of p_n accumulate on some curve, possibly after renormalization? How are the zeroes of p_n related to those p_1? Is there some way in which the zeroes of p_n approach the zeroes of f? Is there a minimum distance between the zeroes of p_n, which is independent of n? In the case f(x) = exp(x) this has been investigated and has led to some nice results.

2) Which Cauchy sequences {a(n)}_n>1 with a(n) > 0, n=1,2,... have the property that the sequence {b(n)}_n>1 is also Cauchy, where
    b(1)=a(1), b(2)=a(2)^b(1), ..., b(n+1)=a(n+1)^b(n)?
We note that the case where a(1) = a(2) = a(3) = ... = a > 0 has been solved, and the solution is both surprising and interesting, though the arguments involved use nothing more than elementary calculus.


Combinatorial Matrix Theory Group  Dr. Leslie Hogben, Dr. Adam Berliner, Dr. Travis Peters, Dr. Michael Young, Nathan Warnberg
Minimum rank, maximum nullity, and zero forcing on a graph

The graph of a real symmetric matrix A=[a_ij] has an edge between i and j if and only if a_ij is nonzero.  Finding the maximum multiplicity of eigenvalue 0 among symmetric matrices having a given graph is the same as finding the maximum nullity and is equivalent to finding the minimum rank among symmetric matrices having the given graph.  Initially a subset Z of the vertices of a graph G are colored blue and the remaining vertices are colored white.  The color change rule is that if a blue vertex v has exactly one white neighbor w, then change the color of w to blue.   The set Z is a zero forcing set if after applying the color change rule until no more changes are possible, all the vertices of G are blue.  The zero forcing number is the minimum size of a zero forcing set.  The zero forcing number is an upper bound for the maximum nullity of a graph, and arose independently in the study of control of quantum systems in physics, where it is called graph infection or propagation.   This project will investigate problems related to minimum rank, maximum nullity, and zero forcing number.

Students involved in this project will be part the ISU Combinatorial Matrix Theory Research Group; more information is available on that page. This group regularly publishes its results (see list of papers).

Linear algebra is a prerequisite for this project, graph theory is an advantage, and a strong theoretical mathematics background (usually including abstract algebra or real analysis) is expected.  The software we use is Sage and Mathematica, so knowing one or both of these in advance is helpful, but you can learn one or both of here.

Automated Recursive Projected CS (ReProCS) for Real-time Video Layering and Performance Guarantees  Dr. Namrata Vaswani, Dr. Leslie Hogben, Brian Lois, Kevin Palmowski, Chassidy Bozeman

A large class of video sequences are composed of at least two layers - the foreground, which is a sparse image that often consists of one or more moving objects, and the background, which is a dense image, that is either constant or changes gradually over time and the changes are usually global. Thus the background sequence is well modeled as lying in a low dimensional subspace that can slowly change with time; while the foreground is well modeled as a sparse "outlier" that changes in a correlated fashion over time (e.g., due to objects' motion). Video layering can thus be posed as a robust principal components' analysis (PCA) problem, with the difference that the "outlier" for PCA is also a signal-of-interest and needs to be recovered too. Real-time video layering then becomes a recursive robust PCA problem. We will develop algorithms using a novel approach called Recursive Projected Compressive Sensing (ReProCS) to solve this problem and we will try to bound their performance under practically motivated assumptions. In particular, we will study how to handle temporal correlations in the low-dimensional part (background).

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