In this paper we investigate the relationship between classes of tree-to-tree-series (for short: t-ts) and o-tree-to-tree-series (for short: o-t-ts) transformations computed by restricted deterministic bottom-up weighted tree transducers (for short: deterministic bu-w-tt). Essentially, deterministic bu-w-tt are deterministic bottom-up tree series transducers [EFV02, FV03, ful, FGV04], but the former are de ned over monoids whereas the latter are de ned over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of non-deletion, linearity, totality, and homomorphism [Eng75] can equivalently be de ned for deterministic bu-w-tt. Using well-known results of classical tree transducer theory (cf., e.g., [Eng75, Fül91]) and also new results on deterministic bu-w-tt, we order classes of t-ts and o-t-ts transformations computed by restricted deterministic bu-w-tt by set inclusion. More precisely, for every commutative monoid we completely specify the inclusion relation of the classes of t-ts and o-t-ts transformations for all sensible combinations of restrictions by means of inclusion diagrams.

Market and nonmarket urban forest resource values can be achieved through many cost reductions (e.g., improved air quality, fossil fuels for heating and cooling, stormwater runoff) and increases in tax bases for communities from improved property values. These benefits need to be measured quantitatively so decision makers can understand economic gains or losses provided by street trees. Resource inventories are often undertaken as part of the planning phase in a tree management program. It is a comprehensive assessment that requires an inventory of a community's tree resources and it acts as a fundamental starting point for most urban and community forestry programs. Whether an inventory is an estimate or a complete count, quantitative benefits and costs for urban forestry programs cannot accurately be represented without one. This study provides a new approach to understanding a city’s street tree structure using data from a Light Detection And Ranging (LiDAR) sensor and other publicly available data (e.g., roads, city boundaries, aerial imagery). This was accomplished through feature (e.g., trees, buildings) extraction from LiDAR data to identify individual trees. Feature extraction procedures were used with basic geographic information system (GIS) techniques and LiDAR Analyst to create street tree inventory maps to be used in determining a community’s benefit/cost ratio (BCR) for its urban forest. Only by explaining an urban forest’s structure can dollar values be assigned to street trees. Research was performed with LiDAR data and a sample of ground control trees in Pass Christian, and Hattiesburg, Mississippi, located in the lower U.S. South where many communities have publicly available geospatial data warehouses (e.g., MARIS in Mississippi, ATLAS in Louisiana). Results from each city’s estimated street trees revealed a BCR 3.23:1 and 6.91:1 for Pass Christian and Hattiesburg, respectively. This study validated a regression model for predicting street tree occurrence in cities using LiDAR Analyst and a street sample. Results demonstrated that using LiDAR Analyst as a street tree inventory tool with publicly available LiDAR data and a sample adequately described 88% of a community’s street trees which was used to calculate both market and nonmarket resource values.

In this report, we study the enumeration of spanning trees in graphs, using two methods withinlinear algebra, Kirchhoff’s Matrix Tree Theorem and an alternative method, also referred to asLemma 1, derived by S. Klee and M.T Stamps in [KS20]. Along with introducing preliminary tools from linear algebra, we also study the Laplace matrix ofa graph and use its properties in the two methods to derive combinatorical expressions on spanningtree enumeration of different graph families. We discuss properties of the Laplace matrix obtainedfrom different graph structures, and determine which method is more suitable in each case, withrespect to linear algebra. Specifically, complete graphs, Ferrers graphs and Windmill graphs areconsidered.

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This dissertation addresses the modeling and analysis of a strategic risk management problem via a novel decision tree optimization approach, as well as development of enhanced Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations for solving nonconvex polynomial programming problems, through the generation of valid inequalities and reduced representations, along with the design and implementation of efficient algorithms. We first conduct a quantitative analysis for a strategic risk management problem that involves allocating certain available failure-mitigating and consequence-alleviating resources to reduce the failure probabilities of system safety components and subsequent losses, respectively, together with selecting optimal strategic decision alternatives, in order to minimize the risk or expected loss in the event of a hazardous occurrence. Using a novel decision tree optimization approach to represent the cascading sequences of probabilistic events as controlled by key decisions and investment alternatives, the problem is modeled as a nonconvex mixed-integer 0-1 factorable program. We develop a specialized branch-and-bound algorithm in which lower bounds are computed via tight linear relaxations of the original problem that are constructed by utilizing a polyhedral outer-approximation mechanism in concert with two alternative linearization schemes having different levels of tightness and complexity. We also suggest three alternative branching schemes, each of which is proven to guarantee convergence to a global optimum for the underlying problem. Extensive computational results and sensitivity analyses are presented to provide insights and to demonstrate the efficacy of the proposed algorithm. In particular, our methodology outperformed the commercial software BARON (Version 8.1.5), yielding a more robust performance along with an 89.9% savings in effort on average. Next, we enhance RLT-based LP relaxations for polynomial programming problems by developing two classes of valid inequalities: v-semidefinite cuts and bound-grid-factor constraints. The first of these uses concepts derived from semidefinite programming. Given an RLT relaxation, we impose positive semidefiniteness on suitable dyadic variable-product matrices, and correspondingly derive implied semidefinite cuts. In the case of polynomial programs, there are several possible variants for selecting such dyadic variable-product matrices for imposing positive semidefiniteness restrictions in order to derive implied valid inequalities, which leads to a new class of cutting planes that we call v-semidefinite cuts. We explore various strategies for generating such cuts within the context of an RLT-based branch-and-cut scheme, and exhibit their relative effectiveness towards tightening the RLT relaxations and solving the underlying polynomial programming problems, using a test-bed of randomly generated instances as well as standard problems from the literature. Our results demonstrate that these cutting planes achieve a significant tightening of the lower bound in contrast with using RLT as a stand-alone approach, thereby enabling an appreciable reduction in the overall computational effort, even in comparison with the commercial software BARON. Empirically, our proposed cut-enhanced algorithm reduced the computational effort required by the latter two approaches by 44% and 77%, respectively, over a test-bed of 60 polynomial programming problems. As a second cutting plane strategy, we introduce a new class of bound-grid-factor constraints that can be judiciously used to augment the basic RLT relaxations in order to improve the quality of lower bounds and enhance the performance of global branch-and-bound algorithms. Certain theoretical properties are established that shed light on the effect of these valid inequalities in driving the discrepancies between RLT variables and their associated nonlinear products to zero. To preserve computational expediency while promoting efficiency, we propose certain concurrent and sequential cut generation routines and various grid-factor selection rules. The results indicate a significant tightening of lower bounds, which yields an overall reduction in computational effort of 21% for solving a test-bed of 15 challenging polynomial programming problems to global optimality in comparison with the basic RLT procedure, and over a 100-fold speed-up in comparison with the commercial software BARON. Finally, we explore equivalent, reduced size RLT-based formulations for polynomial programming problems. Utilizing a basis partitioning scheme for an embedded linear equality subsystem, we show that a strict subset of RLT defining equalities imply the remaining ones. Applying this result, we derive significantly reduced RLT representations and develop certain coherent associated branching rules that assure convergence to a global optimum, along with static as well as dynamic basis selection strategies to implement the proposed procedure. In addition, we enhance the RLT relaxations with v-semidefinite cuts, which are empirically shown to further improve the relative performance of the reduced RLT method over the usual RLT approach. Computational results presented using a test-bed of 10 challenging polynomial programs to evaluate the different reduction strategies demonstrate that our superlative proposed approach achieved more than a four-fold improvement in computational effort in comparison with both the commercial software BARON and a recently developed open-source code, Couenne, for solving nonconvex mixed-integer nonlinear programming problems. Moreover, our approach robustly solved all the test cases to global optimality, whereas BARON and Couenne were jointly able to solve only a single instance to optimality within the set computational time limit, having an unresolved average optimality gap of 260% and 437%, respectively, for the other nine instances. This dissertation makes several broader contributions to the field of nonconvex optimization, including factorable, nonlinear mixed-integer programming problems. The proposed decision tree optimization framework can serve as a versatile management tool in the arenas of homeland security and health-care. Furthermore, we have advanced the frontier for tackling formidable nonconvex polynomial programming problems that arise in emerging fields such as signal processing, biomedical engineering, materials science, and risk management. An open-source software using the proposed reduced RLT representations, semidefinite cuts, bound-grid-factor constraints, and range reduction strategies, is currently under preparation. In addition, the different classes of challenging polynomial programming test problems that are utilized in the computational studies conducted in this dissertation have been made available for other researchers via the Web-page http://filebox.vt.edu/users/dalkiran/website/. It is our hope and belief that the modeling and methodological contributions made in this dissertation will serve society in a broader context through the myriad of widespread applications they support. / Ph. D.

We study a generalization of the problem of finding bounds on the number of discrete chains, which itself is a generalization of the Erdős unit distance problem. Given a set of points in Euclidean space and a tree graph consisting of a much smaller number of vertices, we study the maximum possible number of tree graphs which can be represented by a prescribed tree graph. We derive an algorithm for finding tight bounds for this family of problems up to chain bound discrepancy, and give upper and lower bounds in special cases. / Master of Science / We study a generalization of the problem of finding bounds on the number of discrete chains, which itself is a generalization of the Erdős unit distance problem, a famous mathematics problem named after mathematician Paul Erdős. Given a set of points, and a tree graph of a much smaller amount of vertices, we study the maximum possible number of tree graphs which can be represented by a prescribed tree graph. We derive an algorithm for finding tight bounds for this family of problems up to chain bound discrepancy, and give upper and lower bounds in special cases.

There has been growing public concern regarding the threat to people and environment from industrial activities, thus more rigorous regulations. The investigation of almost all the major accidents shows that we could have avoided those tragedies with effective risk analysis and safety management programs. High-quality risk analysis is absolutely necessary for sustainable development. As a powerful and systematic tool, fault tree analysis (FTA) has been adapted to the particular need of chemical process quantitative risk analysis (CPQRA) and found great applications. However, the application of FTA in the chemical process industry (CPI) is limited. One major barrier is the manual synthesis of fault trees. It requires a thorough understanding of the process and is vulnerable to individual subjectivity. The quality of FTA can be highly subjective and variable. The availability of a computer-based FTA methodology will greatly benefit the CPI. The primary objective of this research is to develop a computer-aided fault tree synthesis methodology for CPQRA. The central idea is to capture the cause-and-effect logic around each item of equipment directly into mini fault trees. Special fault tree models have been developed to manage special features. Fault trees created by this method are expected to be concise. A prototype computer program is provided to illustrate the methodology. Ideally, FTA can be standardized through a computer package that reads information contained in process block diagrams and provides automatic aids to assist engineers in generating and analyzing fault trees. Another important issue with regard to QRA is the large uncertainty associated with available failure rate data. In the CPI, the ranges of failure rates observed could be quite wide. Traditional reliability studies using point values of failure rates may result in misleading conclusions. This dissertation discusses the uncertainty with failure rate data and proposes a procedure to deal with data uncertainty in determining safety integrity level (SIL) for a safety instrumented system (SIS). Efforts must be carried out to obtain more accurate values of those data that might actually impact the estimation of SIL. This procedure guides process hazard analysts toward a more accurate SIL estimation and avoids misleading results due to data uncertainty.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 221-235). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 221-235). Also available on the Internet.

In Renaissance culture there was an iconographic and literary language of trees, related to the motif of the tree of life, an ancient symbol of immortality associated with paradise. The properties of trees were used to express a range of ideas, including the death and resurrection of Christ, the fall and regeneration of political regimes, and virtue and vice within the individual soul. The juxtaposition of the tree of knowledge with the tree of life, as motifs of sterility and fertility, expressed aspects of the human condition and constructions of spiritual history and destiny. This thesis explores the language of trees in visual art and a range of English Renaissance texts from the late-sixteenth to the mid-seventeenth century: two plays by Shakespeare, two country-house poems, and a prose treatise on growing fruit-trees. Each of the writers drew on arboreal metaphors and motifs in unique and innovative ways. However there are numerous parallels and connections between the texts, and with contemporary and antecedent visual art, to justify considering these works together. In Shakespeare’s tragedy Titus Andronicus (1594) Lavinia, when she has her hands cut off, is metaphorically described as a tree with lopped branches and linked with the stricken political entity of Rome. Shakespeare evokes the tree of virtue, the classical myth of Daphne, and the arboreal language of virtue and vice. In the late tragicomedy Cymbeline (1610), the king is symbolized in a dream vision as a tree, with its cut branches representing the princes who are initially stolen but then reunited with the king. The tree represents the family tree as well as the political state, two interlinked concepts in the play and in contemporary iconography and ideology. Since Cymbeline’s reign heralded the Nativity, the prophecy of the lopped and regenerated tree invokes the idea of Christ as the tree of life and the fruit of the tree of Jesse. In both plays, Shakespeare’s tree imagery comments on the exercise of political power and the resultant health of the state. Shakespeare’s contemporary Aemilia Lanyer wrote “The Description of Cooke-ham” (1611), part of a published volume of poetry entitled Salve Deus Rex Judaeorum. In the poem she imagines a prominent tree on the estate as the tree of life. An abstract metaphor is envisaged as part of the physical landscape. The motif transforms the estate to sacred terrain, enabling her to claim access to a space she is otherwise excluded from by class and gender. Lanyer links the sap from the tree of life with her writing, seeking to legitimize her claim as a female poet. Such strategies are part of her bid for patronage from the Countess of Cumberland, her primary dedicatee. In another country-house poem, Andrew Marvell’s “Upon Appleton House, to my Lord Fairfax” (1651), the poet creates a forest of the mind in which he explores different aspects of the language of trees. The speaker imagines himself encircled by vines and crucified by thorns, in imitation of Christ as the tree of life, while a fallen oak tree suggests the regicide. He takes on various roles including that of the enigmatic Green Man. I place Marvell’s imagery in the context of the Civil War and the relationship with his employer Lord Fairfax. Marvell’s exploration of arboreal motifs also subjects Christian tree of life imagery to the challenge of its pagan antecedents and reflects anxieties over the natural processes that threaten metaphors of regeneration. Lastly, in Ralph Austen’s A Treatise of Fruit-trees and Spiritual Use of an Orchard (1653), the author blends advice on horticultural practices in growing fruit-trees with religious metaphors. For Austen, gardening is both a physical and a metaphysical pursuit. His readers are expected to plant fruit-trees in orchards that evoke the idea of Christ as the tree of life and related ideas. His use of the motif is part of his advocacy of agricultural and social reform, motivations that were part of those in the circle surrounding Samuel Hartlib. Austen’s text is situated at the end of the English Renaissance and at the beginnings of the Scientific Revolution, when emblematic and symbolic frameworks for interpreting the natural world were subject to new pressures derived from empirical and rationalistic outlooks. What becomes apparent from these works is that tree metaphors were literalized, just as they had been in visual art, and given a new naturalism as they were projected onto landscapes. Symbolic trees merged with botanical trees in imagined landscapes, creating hybrid terrains that were both descriptive and mythical. Recognition of the language of trees in Renaissance culture opens up new readings of both canonical and lesser-known texts and highlights the porous disciplinary border between literature and art. Our historical readings are richer for understanding the potent language of trees. Overall the thesis highlights the importance and cultural preoccupation with trees in European visual and literary traditions.