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

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Teacher Training Syllabus: Mathematics for Adults

Teacher Training Syllabus in Mathematics for Adults written by Lyle Leland in 1971

Parameter and Structure Learning Techniques for Sum Product Networks

Probabilistic graphical models (PGMs) provide a general and flexible framework for reasoning about complex dependencies in noisy domains with many variables. Among the various types of PGMs, sum-product networks (SPNs) have recently generated some interest because exact inference can always be done in linear time with respect to the size of the network. This is particularly attractive since it means that learning an SPN from data always yields a tractable model for inference. Learning the parameters and the structure for SPNs is being explored by various researchers, having algorithms that scale are essential in the era of big data. In this thesis, I present tractable parameter and structure learning techniques for SPNs. First, I propose a new Bayesian moment matching (BMM) algorithm to learn the parameters for SPNs generatively. BMM operates naturally in an online fashion and that can be easily distributed. I demonstrate the effectiveness and scalability of BMM in comparison to other online algorithms in the literature. Second, I present a discriminative learning algorithm for SPNs based on the Extended Baum-Welch (EBW) algorithm. The experiments show that this algorithm performs better than both generative Expectation-Maximization, and discriminative gradient descent on a wide variety of applications. I also demonstrate the robustness of the algorithm in the case of missing features by comparing its performance to Support Vector Machines and Neural Networks. Finally, I present the first online structure learning algorithm for recurrent SPNs. Recurrent SPNs were proposed by Mazen et. al to model sequential data. They also proposed a structure learning algorithm which is slow, and it only operates in batch mode. I present the first online algorithm to learn the structure of recurrent SPNs. I also show how the parameters can be learned simultaneously using a modified version of hard-EM algorithm. I compare the performance of the algorithm against different models on sequential data problems

Solving polynomial constraints for proving termination of rewriting

A termination problem can be transformed into a set of polynomial constraints. Up to now, several approaches have been studied to deal with these constraints as constraint solving problems. In this thesis, we study in depth some of these approaches, present some advances in each approach.Navarro Marset, RA. (2008). Solving polynomial constraints for proving termination of rewriting. http://hdl.handle.net/10251/13626Archivo delegad