Branched Polymers and Hyperplane Arrangements

Original manuscript December 17, 2009We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie (Ann Math, 158:1019–1039, 2003), and Kenyon and Winkler (Am Math Mon, 116(7):612–628, 2009) to any central hyperplane arrangement A A . The volume of the resulting configuration space of connected branched polymers associated to the hyperplane arrangement A A is expressed through the value of the characteristic polynomial of A A at 0. We give a more general definition of the space of branched polymers, where we do not require connectivity, and introduce the notion of q-volume for it, which is expressed through the value of the characteristic polynomial of A A at −q − q . Finally, we relate the volume of the space of branched polymers to broken circuits and show that the cohomology ring of the space of branched polymers is isomorphic to the Orlik–Solomon algebra.National Science Foundation (U.S.) (Grant DMS 6923772)National Science Foundation (U.S.) (CAREER Award DMS 0504629

Enumerating Acyclic Orientations

An acyclic orientation (AO) of an undirected graph is an assignment of direction to each of its edges without introducing a directed cycle. We study enumeration problems regarding AOs. Our results include: an explicit formula for the the number of AOs of 3xn grid graphs and complete multipartite graphs, which answers a question raised by Cameron, Glass, and Schumacher (2014). We also provide a bijection between AOs of complete bipartite graphs with a fixed unique sink vertex and permutations with a prescribed excedance set, relating two combinatorial objects not previously known to be connected. Finally, we consider Markov chains on AOs for the purpose of efficiently sampling a random uniform AO from any graph. Enumerating AOs is of interest for the connections they share with graph coloring. Indeed, the number of AOs is given by the chromatic polynomial evaluation at -1. This enumeration problem is also studied in computer science as a #P-complete Tutte polynomial evaluation with unknown approximability. Finally, it appears in biology as the enumeration of branched polymers, and statistical physics as the Ursell function.Undergraduat

Characteristic Polynomial of Arrangements and Multiarrangements

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck. In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction. In chapter 3, we use some tools from the intersection theory and show that the intersection cycle of a certain projective variety has a closed answer in terms of the characteristic polynomial. This result is used to compute the leading parts of the Hilbert polynomial and Hilbert series of the logarithmic ideal. As a consequence, we recover some of the classical results of the theory such as the Solomon-Terao formula for tame arrangements. This is done by computing the Hilbert series in two different ways. We also introduce the notion of logarithmic Orlik-Terao ideal and show that the intersection lattice parametrizes a primary decomposition. The chapter is closed by a generalization of logarithmic ideals to higher orders. It is shown that these ideals detect the freeness of the corresponding logarithmic modules. The last chapter is a generalization of the notion of logarithmic ideal to multiarrangements. Some of the basic properties of these ideals are investigated. It is shown that one obtains a natural resolution of this ideal by logarithmic modules under the tame condition. In the final section it is shown that the intersection cycle of the logarithmic ideal of a free multiarrangement is obtained from its characteristic polynomial, similar to simple arrangements

Machine Learning Assisted Discovery of Shape Memory Polymers and Their Thermomechanical Modeling

As a new class of smart materials, shape memory polymer (SMP) is gaining great attention in both academia and industry. One challenge is that the chemical space is huge, while the human intelligence is limited, so that discovery of new SMPs becomes more and more difficult. In this dissertation, by adopting a series of machine learning (ML) methods, two frameworks are established for discovering new thermoset shape memory polymers (TSMPs). Specifically, one of them is performed by a combination of four methods, i.e., the most recently proposed linear notation BigSMILES, supplementing existing dataset by reasonable approximation, a mixed dimension (1D and 2D) input model and a type of dual-convolutional-neural-network model. The framework is able to predict the recovery stress and glass transition temperature for TSMP and screen 14 new TSMPs from a large chemical space. The other leverages transfer learning, variational autoencoder and weighted vector combination method, and the developed ML framework can design ultraviolet (UV) curable TSMPs with desired properties. With new SMPs discovered by ML, as well as other new SMPs continuously developed in the labs, there is an urgent need to develop thermomechanical models so that the new SMPs can be used in structural design. Through the framework of solid mechanics, three different constitutive models are presented for classical one-way thermoset shape memory polymer (TSMP), two-way semi-crystalline SMP and enthalpy-driven four-chain SMP with large recovery stress, respectively. Among them, a new two-phase sphere model based on the physical growth process of the frozen phase from nuclei is proposed, which tends to bring more underly physical mechanism for the classical storage strain-based phase transition model. By introducing Gibbs energy and a transition of the molecule deformation mechanism, a enthalpy-driven thermomechanical model with new representative unit cell is developed, which could reasonably elucidates the large recovery stress for a new branch of TSMPs. Multiple mechanisms, involving phase transition law, damage evolution, and relaxation are introduced into the model for two-way semi-crystalline SMP, which is able to reveal the mechanisms of three different 2W-SMEs

The Use of Computational Methods in the Grouping and Assessment of Chemicals - Preliminary Investigations

This document presents a perspective of how computational approaches could potentially be used in the grouping and assessment of chemicals, and especially in the application of read-across and the development of chemical categories. The perspective is based on experience gained by the authors during 2006 and 2007, when the Joint Research Centre's European Chemicals Bureau was directly involved in the drafting of technical guidance on the applicability of computational methods under REACH. Some of the experience gained and ideas developed resulted from a number of research-based case studies conducted in-house during 2006 and the first half of 2007. The case studies were performed to explore the possible applications of computational methods in the assessment of chemicals and to contribute to the development of technical guidance. Not all of the methods explored and ideas developed are explicitly included in the final guidance documentation for REACH. Many of the methods are novel, and are still being refined and assessed by the scientific community. At present, many of the methods have not been tried and tested in the regulatory context. The authors therefore hope that the perspective and case studies compiled in this document, whilst not intended to serve as guidance, will nevertheless provide an input to further research efforts aimed at developing computational methods, and at exploring their potential applicability in regulatory assessment of chemicals.JRC.I.3-Toxicology and chemical substance

Selected Works in Bioinformatics

This book consists of nine chapters covering a variety of bioinformatics subjects, ranging from database resources for protein allergens, unravelling genetic determinants of complex disorders, characterization and prediction of regulatory motifs, computational methods for identifying the best classifiers and key disease genes in large-scale transcriptomic and proteomic experiments, functional characterization of inherently unfolded proteins/regions, protein interaction networks and flexible protein-protein docking. The computational algorithms are in general presented in a way that is accessible to advanced undergraduate students, graduate students and researchers in molecular biology and genetics. The book should also serve as stepping stones for mathematicians, biostatisticians, and computational scientists to cross their academic boundaries into the dynamic and ever-expanding field of bioinformatics