Normal Domains Arising from Graph Theory

Determining whether an arbitrary subring R of k[x1±1,...,xn±1] is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. First, we determine normality in the case where R is a monomial generated domain where the generators have the form (xixj)±1. Using results for this special case we generalize to the case when R is a monomial generated domain where the generators have the form xi±1xj±1. In both cases, for the ring R, we consider the combinatorial structure that assigns an edge in a mixed directed signed graph to each monomial of the ring. We then use this relationship to provide a combinatorial characterization of the normality of R, and, when R is not normal, we use the combinatorial characterization to compute the normalization of R. Using this construction, we also determine when the ring R satisfies Serre\u27s R1 condition. We also discuss generalizations of this to directed graphs with a homogenizing variable and a special class of hypergraphs

2011 Undergraduate Research Symposium Abstract Book

Abstract book from the 2011 Eleventh Annual UMM Undergraduate Research Symposium (URS) which celebrates student scholarly achievement and creative activities

Fast Parallel Algorithms on a Class of Graph Structures With Applications in Relational Databases and Computer Networks.

The quest for efficient parallel algorithms for graph related problems necessitates not only fast computational schemes but also requires insights into their inherent structures that lend themselves to elegant problem solving methods. Towards this objective efficient parallel algorithms on a class of hypergraphs called acyclic hypergraphs and directed hypergraphs are developed in this thesis. Acyclic hypergraphs are precisely chordal graphs and their subclasses, and they have applications in relational databases and computer networks. In this thesis, first, we present efficient parallel algorithms for the following problems on graphs. (1) determining whether a graph is strongly chordal, ptolemaic, or a block graph. If the graph is strongly chordal, determine the strongly perfect vertex elimination ordering. (2) determining the minimal set of edges needed to make an arbitrary graph strongly chordal, ptolemaic, or a block graph. (3) determining the minimum cardinality dominating set, connected dominating set, total dominating set, and the domatic number of a strongly chordal graph. Secondly, we show that the query implication problem (Q\sb1\ \to\ Q\sb2) on two queries, which is to determine whether the data retrieved by query Q\sb1 is always a subset of the data retrieved by query Q\sb2, is not even in NP and in fact complete in \Pi\sb2\sp{p}. We present several \u27fine-grain\u27 analyses of the query implication problem and show that the query implication can be solved in polynomial time given chordal queries. Thirdly, we develop efficient parallel algorithms for manipulating directed hypergraphs H such as finding a directed path in H, closure of H, and minimum equivalent hypergraph of H. We show that finding a directed path in a directed hypergraph is inherently sequential. For directed hypergraphs with fixed degree and diameter we present NC algorithms for manipulations. Directed hypergraphs are representation schemes for functional dependencies in relational databases. Finally, we also present an efficient parallel algorithm for multi-dimensional range search. We show that a set of points in a rectangular parallelepiped can be obtained in O(logn) time with only 2.log\sp2 n $-$ 10.logn + 14 processors on a EREW-PRAM. A nontrivial implementation technique on the hypercube parallel architecture is also presented. Our method can be easily generalized to the case of d-dimensional range search

2007 Undergraduate Research Symposium Abstract Book

Abstract book from the 2007 UMM Undergraduate Research Symposium (URS) which celebrates student scholarly achievement and creative activities

Structured Representations for Coreference Resolution

Coreference resolution is the task of determining which expressions in a text are used to refer to the same entity. This task is one of the most fundamental problems of natural language understanding. Inherently, coreference resolution is a structured task, as the output consists of sets of coreferring expressions. This complex structure poses several challenges since it is not clear how to account for the structure in terms of error analysis and representation. In this thesis, we present a treatment of computational coreference resolution that accounts for the structure. Our treatment encompasses error analysis and the representation of approaches to coreference resolution. In particular, we propose two frameworks in this thesis. The first framework deals with error analysis. We gather requirements for an appropriate error analysis method and devise a framework that considers a structured graph-based representation of the reference annotation and the system output. Error extraction is performed by constructing linguistically motivated or data-driven spanning trees for the graph-based coreference representations. The second framework concerns the representation of approaches to coreference resolution. We show that approaches to coreference resolution can be understood as predictors of latent structures that are not annotated in the data. From these latent structures, the final output is derived during a post-processing step. We devise a machine learning framework for coreference resolution based on this insight. In this framework, we have a unified representation of approaches to coreference resolution. Individual approaches can be expressed as instantiations of a generic approach. We express many approaches from the literature as well as novel variants in our framework, ranging from simple pairwise classification approaches to complex entity-centric models. Using the uniform representation, we are able to analyze differences and similarities between the models transparently and in detail. Finally, we employ the error analysis framework to perform a qualitative analysis of differences in error profiles of the models on a benchmark dataset. We trace back differences in the error profiles to differences in the representation. Our analysis shows that a mention ranking model and a tree-based mention-entity model with left-to-right inference have the highest performance. We discuss reasons for the improved performance and analyze why more advanced approaches modeled in our framework cannot improve on these models. An implementation of the frameworks discussed in this thesis is publicly available

Proceedings of the 1994 Monterey Workshop, Increasing the Practical Impact of Formal Methods for Computer-Aided Software Development: Evolution Control for Large Software Systems Techniques for Integrating Software Development Environments

Office of Naval Research, Advanced Research Projects Agency, Air Force Office of Scientific Research, Army Research Office, Naval Postgraduate School, National Science Foundatio

A Visual Language for Composable Simulation Scenarios

Modeling and Simulation plays an important role in how the Air Force trains and fights, Scenarios are used in simulation to give users the ability to specify entities and behaviors that should be simulated by a model: however, building and understanding scenarios can be a difficult and time-consuming process, furthermore, as composable simulations become more prominent, the need for a common descriptor for simulation scenarios has become evident. In order to reduce the complexity of creating and understanding simulation scenarios, a visual language was created, The research on visual languages presented in this thesis examines methods of visually specifying the high-level behavior of entities in scenarios and how to represent the hierarchy of the entities in scenarios. Through a study of current behavior specification techniques and the properties of mission-level simulation scenarios, Simulation Behavior Specification Diagrams (SBSD) were developed to represent the behavior of entities in scenarios, Additionally, the information visualization technique of treemaps was adapted to represent the hierarchy of entities in scenarios, After completing case studies on scenarios for the OneSAF simulation model, SBSDs and the application of treemaps to scenarios was considered successful, SBSD diagrams accurately represented the behavior of entities in the simulation scenarios and through software can be converted into code for use by simulation models, The treemap displayed the hierarchy of the entities along with information about the relative size of the entities when applied to simulation scenarios