Advisers: Richard Rebarber and Brigitte Tenhumberg

]]>Advisors: Mark Brittenham and Susan Hermiller

]]>Advisors: Mark Brittenham and John Meakin

]]>Vision plays an important role in how we interact with our environments. To fully understand how visual information is processed requires an understanding of the way signals are transformed at the very first synapse: the ribbon synapse of photoreceptor neurons (rods and cones). These synapses possess a ribbon-like structure on which approximately 100 synaptic vesicles can be stored, allowing graded responses through the release of different numbers of vesicles in response to visual input. These responses depend critically on the ability of the ribbon to replenish itself as ribbon sites empty upon release. The rate of vesicle replenishment is thus an important factor in shaping neural coding in the retina. In collaboration with experimental neuroscientists we developed a mathematical model to describe the dynamics of vesicle release and replenishment at the ribbon synapse.

To learn more about how network architecture shapes the dynamics of the network, we study a specific type of threshold-linear network that is constructed from a simple directed graph. These networks are particularly well suited for our study because the network construction guarantees that differences in dynamics arise solely from differences in the connectivity of the underlying graph. By design, the activity of these networks is bounded and there are no stable fixed points. Computational experiments show that most of these networks yield limit cycles where the neurons fire in sequence. Can we predict the order in which the neurons fire? To this end, we devised an algorithm to predict the sequence of firing using the structure of the underlying graph. Using the algorithm we classify all the networks of this type on five or fewer nodes.

Adviser: Carina Curto

]]>Adviser: Allan C. Peterson

]]>This work is motivated by Glaz's question regarding whether a notion of Cohen-Macaulay exists for coherent rings which satisfies certain properties and agrees with the usual notion when the ring is Noetherian. Hamilton and Marley gave one answer; we develop an alternative approach using homological dimensions which seems to have more satisfactory properties. We explore properties of coherent Cohen-Macaulay rings, as well as their relationship to non-Noetherian Cohen-Macaulay rings as defined by Hamilton and Marley.

Adviser: Tom Marley

]]>Advisers: Luchezar L. Avramov and Srikanth B. Iyengar

]]>In Part I, we use complete (injective) resolutions to define a stable version of local cohomology. For a module having a complete injective resolution, we associate a stable local cohomology module; this gives a functor to the stable category of Gorenstein injective modules. We show that this functor behaves much like the usual local cohomology functor. When there is only one non-zero local cohomology module, we show there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.

In Part II, we utilize minimal cotorsion-flat resolutions to compute cosupport. We first develop a criterion for a cotorsion-flat resolution to be minimal. For a module having an appropriately minimal resolution by cotorsion-flat modules, we show that its cosupport coincides with those primes ``appearing'' in such a resolution---much like the dual notion that minimal injective resolutions detect (small) support. This gives us a method to compute the cosupport of various modules, including all flat modules and all cotorsion modules.

Adviser: Mark E. Walker

]]>Adviser: Thomas Marley

]]>We establish basic mathematical properties of bias matrices, in particular describing the fundamental polytope in which all biases reside (Theorem 3.25). We show how the underlying simple digraph of a bias matrix, which we term the bias network, serves as a combinatorial generalization of neuronal templates. Surprisingly, every simple digraph is realizable as the bias network of some sequence (Theorem 3.34). The bias-matrix representation leads us to a natural method for sequence correlation, which then leads to a probabilistic framework for determining the similarity of one set of sequences to another. Using data from rat hippocampus, we describe events of interest and also sequence-detection techniques. Finally, the bias matrix and the similarity measure are applied to this real-world data using code developed by us.

Advisor: Vladimir Itskov

]]>Adviser: Mark Brittenham and Susan Hermiller

]]>Adviser: Mark Brittenham

]]>Advisers: Petronela Radu and Daniel Toundykov

]]>The *degree sequence* of a (hyper)graph is the list of the number of edges containing each vertex. A *t-switch* replaces* t* edges with* t* new edges while maintaining the same degree sequence. For graphs, it has been repeatedly shown that any realization of a degree sequence can be turned into any other realization by a sequence of 2-switches. However, Gabelman provided an example to show 2-switches are not sufficient for *k*-graphs with *k* ≥ 3. We classify all pairs of 3-graphs that do not admit a 2-switch but differ by a 3-switch. We use this to provide support that 2-switches and a 3-switch are sufficient for 3-graphs.

Given graphs *G* and *H*, *G* is *H-saturated* if *G* does not contain *H* as a subgraph, but *H* is a subgraph of *G+e* for any *e* not in *E*(*G*). While this is well defined for subgraphs, the similar definition is not well defined for induced subgraphs. To avoid this, Martin and Smith defined the* induced-saturation number* using trigraphs. We show that the induced-saturation number of stars is zero. This implies the existence of graphs that are star induced-saturated. We introduce the parameter indsat*(*n,H*) which is the minimum number of edges in an *H*-induced-saturated graph, when one exists. We provide bounds for indsat*(*n,K*_{1,3}) and compute it exactly for infinitely many *n*.

Adviser: Stephen Hartke

]]>Advisers: Mark Brittenham and Susan Hermiller

]]>Advisers: Bo Deng and Etsuko Moriyama

]]>1) Polynomial growth of Betti sequences over local rings (Chapter 2),

2) Connected sums of Gorenstein rings (Chapter 3).

Chapter 1 gives an introduction for the two topics discussed in this thesis.

The first part of the thesis deals with modules over complete intersections using free resolutions. The asymptotic patterns of the Betti sequences of the finitely generated modules over a local ring ** R** reflect and affect the singularity of

The second part of the thesis studies a construction on the set of Gorenstein local rings, known as their connected sum. Given a Gorenstein ring, one would like to know whether it can be decomposed as a connected sum and if so, what are its components. We give a concrete description in the case of Gorenstein Artin local algebra over a field. We further investigate conditions on the decomposability of some classes of Gorenstein Artin rings. This is joint work with Hariharan and Celikbas.

Adviser: Luchezar Avramov

]]>We first observe in the preliminaries chapter that for graphs with a fixed number of vertices and edges there is a threshold graph attaining the minimum number of matchings. The first two major results develop this fact in two different directions. In Chapter 3 we consider the problem of maximizing the number of matchings in the class of threshold graphs. We solve the problem completely, concluding that a graph in this class has the maximum number of matchings if and only if it is almost alternating. The second and more fundamental question is the problem of which threshold graph, and hence which graph, has the minimum number of matchings. Ahlswede and Katona determined which graph has the fewest matchings of size 2. In Chapter 4 we extend this result to all sizes of matchings and to the total number of matchings. We prove that either the lex graph or the colex graph minimizes the number of matchings. We further prove that the lex bipartite graph has the fewest matchings among all bipartite graphs with parts of fixed sizes.

Finally, in Chapter 5, we answer an extremal question about independent sets in hypergraphs. ... The final chapter discusses results about maximizing *s*-independent sets in *r*-uniform hypergraphs, the most significant result finds the 3-uniform hypergraph maximizing the number of 2-independent sets for certain numbers of vertices and edges.

Adviser: Jamie Radcliffe

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