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Pages
 Title
 HinduArabic numerals.
 Creator

Maddox, Russell C., Wade, T. L., Florida State University
 Abstract/Description

"Each of the ancient nations had a number system of their own, however, only a few of these will be mentioned in this paper, those being the ones which probably contributed to the development of our own number system. Therefore, the primary purpose of this paper is to give a gist of these ancient number systems, and to give in more details the history of our present system, namely, the HinduArabic number system. Although this is not a complete history of our system of numbers, it is hoped...
Show more"Each of the ancient nations had a number system of their own, however, only a few of these will be mentioned in this paper, those being the ones which probably contributed to the development of our own number system. Therefore, the primary purpose of this paper is to give a gist of these ancient number systems, and to give in more details the history of our present system, namely, the HinduArabic number system. Although this is not a complete history of our system of numbers, it is hoped that the reader will see some advantages the system has over the number systems of the ancient world"Introduction.
Show less  Date Issued
 1945
 Identifier
 FSU_historic_aku8685
 Format
 Thesis
 Title
 The value of socialized arithmetic drills and tests as positive factors in personality development.
 Creator

Rhoads, Louise Nash, Edwards, W., Florida State University
 Abstract/Description

"This paper is an account of a project which began primarily as an attempt to set up an activity program that would result in a high degree of accuracy in computational operations. It soon became evident that the activity was serving as an instrument for removing emotional blocks and relieving frustrations. A pleasant atmosphere of cooperative interprise took the place of classroom discipline, and the teacher found herself no longer a 'party to,' but a partner in a learning situation. The...
Show more"This paper is an account of a project which began primarily as an attempt to set up an activity program that would result in a high degree of accuracy in computational operations. It soon became evident that the activity was serving as an instrument for removing emotional blocks and relieving frustrations. A pleasant atmosphere of cooperative interprise took the place of classroom discipline, and the teacher found herself no longer a 'party to,' but a partner in a learning situation. The program has completed its fourth year. It has been revised, adapted, and added to, when the situation indicated a need. The classes taking part in the most recently revised activities are the ones whose performances are described in this paper"Introduction.
Show less  Date Issued
 1949
 Identifier
 FSU_historic_AKP4901
 Format
 Thesis
 Title
 How can the mathematics teacher interpret "meaning"?.
 Creator

Perfect, Mary, Trimble, H. C., Florida State University
 Abstract/Description

"The fundamental aim of mathematics instruction is the teaching of problem solving. For the majority of people this means mathematics is a language with which they can express quantitative relationships. The meanings in this language must be understood; the language must be practiced and applied to life situations if mathematics is to serve its purpose in the school curriculum. To teach mathematics in this manner is a tremendous job. It requires a teacher not only with patience, understanding...
Show more"The fundamental aim of mathematics instruction is the teaching of problem solving. For the majority of people this means mathematics is a language with which they can express quantitative relationships. The meanings in this language must be understood; the language must be practiced and applied to life situations if mathematics is to serve its purpose in the school curriculum. To teach mathematics in this manner is a tremendous job. It requires a teacher not only with patience, understanding of the students, and an excellent mathematics background but also with a broader general background. He will need to be able to find varied functional relationships of mathematics to life and must have the skill to help the students understand these relationships, too. In summary, good mathematics instruction includes the proper proportions and the proper interweaving of meanings, drill, and applications at the appropriate level of the student"Introduction.
Show less  Date Issued
 1949
 Identifier
 FSU_historic_akp4902
 Format
 Thesis
 Title
 The mathematics teacher uses sports.
 Creator

Woolsey, Kenneth V., Florida State University
 Abstract/Description

What can be done to bring secondary mathematics courses in tempo with the present day needs and interests of the student? The purpose of this paper is to suggest a partial answer to this question. It is doubtless true that most boys and girls in the secondary school are far more interested in sports than in mathematics. Why not draw upon this common interest and bring sports into the mathematics classroomor even take the mathematics classroom out to the field of sports? Such a question may...
Show moreWhat can be done to bring secondary mathematics courses in tempo with the present day needs and interests of the student? The purpose of this paper is to suggest a partial answer to this question. It is doubtless true that most boys and girls in the secondary school are far more interested in sports than in mathematics. Why not draw upon this common interest and bring sports into the mathematics classroomor even take the mathematics classroom out to the field of sports? Such a question may seem unreasonable to those who have not given much thought to the possibility of approaching certain phases of mathematics through student interests in sports. Actually, such an approach is not at all unreasonable. The sports world offers practical examples of numerous mathematical relationships.
Show less  Date Issued
 1949
 Identifier
 FSU_historic_akp2736
 Format
 Thesis
 Title
 A study of the general mathematics program in the secondary school.
 Creator

Cannon, Ruby R., Curtis, H. A., Florida State University
 Abstract/Description

This paper is a study of the general mathematics program in the secondary school. The purpose of this study is to help the inexperienced teacher plan his program of work for the year. The teacher will not find a definite program that he may follow stepbystep, but suggestions that will be of help in developing an effective program.
 Date Issued
 1950
 Identifier
 FSU_historic_akv1201
 Format
 Thesis
 Title
 Further analysis of test data: Basic mathematics for general education.
 Creator

Blake, John W., Florida State University
 Abstract/Description

A mass of raw data is available in the form of test scores of students who have completed "Basic mathematics for general education". These data have been only partially analyzed. This paper records some efforts to decide what statistical techniques might help one to see more in this mass of data to understand more fully the implications of these data. A better understanding of these data is of vital interest to the teaching staff of "Basic mathematics" in their efforts to improve the course.
 Date Issued
 1950
 Identifier
 FSU_historic_akz5817
 Format
 Thesis
 Title
 The language of mathematics.
 Creator

Kmen, Henry Arnold, Florida State University
 Abstract/Description

In the catalog issue of the Bulletin of Florida State University, to be published sometime during 1950, will appear a challenging innovation: the General Education course in mathematics, titled mathematics 105, will be listed under the general area of Communication through Language. This, so far as the writer is able to ascertain, will be the first time a course in mathematics has been so listed in any university catalog. It will be the purpose of this paper to examine some aspects of the...
Show moreIn the catalog issue of the Bulletin of Florida State University, to be published sometime during 1950, will appear a challenging innovation: the General Education course in mathematics, titled mathematics 105, will be listed under the general area of Communication through Language. This, so far as the writer is able to ascertain, will be the first time a course in mathematics has been so listed in any university catalog. It will be the purpose of this paper to examine some aspects of the historical development of mathematics to justify such a classification and to explore some of the implications of such an approach for the teaching of mathematics.
Show less  Date Issued
 1950
 Identifier
 FSU_historic_akw1893
 Format
 Thesis
 Title
 A proposed plan for guiding learning experiences of eighth grade pupils in mathematics.
 Creator

Harrison, Estel Wade, Edwards, W., Florida State University
 Abstract/Description

"Out of genuine desire to prepare oneself to handle, in a more effective way, the teaching of eighth grade mathematics, there comes to mind such questions as these: 1. What are the needs or tasks or problems of eighth grade pupils to which arithmetic can make a contribution? 2. What content is available in the state adopted textbooks? 3. How well is this material adapted to school needs of pupils of this age? 4. What reliable tests can be found? 5. What materials and plans of a general nature...
Show more"Out of genuine desire to prepare oneself to handle, in a more effective way, the teaching of eighth grade mathematics, there comes to mind such questions as these: 1. What are the needs or tasks or problems of eighth grade pupils to which arithmetic can make a contribution? 2. What content is available in the state adopted textbooks? 3. How well is this material adapted to school needs of pupils of this age? 4. What reliable tests can be found? 5. What materials and plans of a general nature can be found or developed which, if revised later to fit the specific classroom situation, may prove of help in improving the teaching of mathematics?"Introduction.
Show less  Date Issued
 1952
 Identifier
 FSU_historic_AKP4877
 Format
 Thesis
 Title
 Making arithmetic meaningful to young children.
 Creator

Thomas, Ethel Paderick Rosasco, Swearingen, Mildred E., Florida State University
 Abstract/Description

"Wanting to help children to overcome any fears that might be foremost in them, the writer wishes to make a study of principles of teaching arithmetic and apply in the classroom certain of these principles in an effort to help children hurdle their great fear of arithmetic"Introduction.
 Date Issued
 1956
 Identifier
 FSU_historic_AKP4832
 Format
 Thesis
 Title
 Distance spaces.
 Creator

Willcox, Herbert S., Griffith, H. C., Florida State University
 Abstract/Description

"The purpose of this paper is to record the results of a study of an abstract set upon which a distance function, having certain properties, has been defined. It is assumed that the reader is familiar with the fundamental concepts of set theory"Introduction.
 Date Issued
 1959
 Identifier
 FSU_historic_AKP2815
 Format
 Thesis
 Title
 A proposed plan for the use of the mathematics portion of the ninth grade testing program.
 Creator

Lane, Rodney A., Schultz, Raymond E., Florida State University
 Abstract/Description

There is in effect at the present time a Florida StateWide NinthGrade Testing Program. This program consists of two tests; one is the School Ability Test (SAT) and the other consists of five parts of the Iowa Tests of Educational Development (ITED). The SAT is an attempt to provide a measure of a student's ability to achieve successfully in a school program. This test provides two part scores, verbal and quantitative, and a total score which is the sum of the part scores. The quantitative...
Show moreThere is in effect at the present time a Florida StateWide NinthGrade Testing Program. This program consists of two tests; one is the School Ability Test (SAT) and the other consists of five parts of the Iowa Tests of Educational Development (ITED). The SAT is an attempt to provide a measure of a student's ability to achieve successfully in a school program. This test provides two part scores, verbal and quantitative, and a total score which is the sum of the part scores. The quantitative section of the test is designed to give "measure of ability in certain quantitative skills of number manipulation and problem solving." Items on this test are of two kinds; one involves problem solving and the other involves numerical computation. When the SAT is mentioned in this paper, it is the quantitative section that is referred to.
Show less  Date Issued
 1960
 Identifier
 FSU_historic_akw1838
 Format
 Thesis
 Title
 Inductive discovery learning, reception learning, and formal verbalization of mathematical concepts.
 Creator

Hanson, Lawrence Eugene, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

Theoretical speculations abound on all sides of the following two questions: 1. What are the relative merits of the reception and discovery modes of learning? 2. What effect does forcing a student to immediately verbalize his newly discovered concept have on his ability to retain and transfer this concept? The purpose of the present study is to seek answers to these questions on the basis of experimental evidence.
 Date Issued
 1967
 Identifier
 FSU_agh3055
 Format
 Thesis
 Title
 The development and testing of a teachtest instrument for prediction of success in college freshman mathematics.
 Creator

Smith, Joe Kelly, Heimer, Ralph T., Florida State University
 Abstract/Description

"The purpose of this research is the development and testing of an instrument to be used in prediction of success in college freshman mathematics courses"Introduction.
 Date Issued
 1967
 Identifier
 FSU_ahm6748
 Format
 Thesis
 Title
 A study of interactions between "StructureofIntellect" factors and two methods of presenting concepts of modulus seven arithemetic.
 Creator

Behr, Merlyn J., Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"In general terms, the purposes of this study were two in number: (1) to suggest whether unique mental factors as identified by methods of factor analysis are correlated with success in usual school learning situations and (2) to suggest whether it is possible to design instructional materials in a way which would suit the learner's mental ability profile"Introduction.
 Date Issued
 1967
 Identifier
 FSU_ahp9230
 Format
 Thesis
 Title
 The effect of the knowledge of logic in proving mathematical theorems in the context of mathematical induction.
 Creator

Walter, Robert Lee, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"Let P(n) be a statement for every positive integer n. We denote the set of all positive integers by N and consider G = {n [is an element of] N [such that] P(n) is true}. The principle of mathematical induction can now be stated as follows: If [(i) 1 [is an element of] G and, (ii) for all k [is an element of] N if k [is an element of] G, then k + 1 [is an element of] G], then G = N. Now symbolize this statement as follows: P: 1 [is an element of] G. R: k [is an element of] G. S: k + 1 [is an...
Show more"Let P(n) be a statement for every positive integer n. We denote the set of all positive integers by N and consider G = {n [is an element of] N [such that] P(n) is true}. The principle of mathematical induction can now be stated as follows: If [(i) 1 [is an element of] G and, (ii) for all k [is an element of] N if k [is an element of] G, then k + 1 [is an element of] G], then G = N. Now symbolize this statement as follows: P: 1 [is an element of] G. R: k [is an element of] G. S: k + 1 [is an element of] G. Q: G = N. Therefore the statement of the principle of mathematical induction can be seen in the following form. If [P and, [for all] k [is an element of] N (if R, then S)], then Q. One strategy for teaching this principle is to explain that in order to apply the principle of mathematical induction and assert Q, one must appeal to the logical rule of modus ponens (the law of detachment). That is, we must affirm the antecedent [P and, [for all] k [is an element of] N (if R, then S)], and then we can assert Q. Therefore the research hypothesis for this study was that if people have the prerequisite knowledge of logic, and that if they are taught the principle of mathematical induction in terms of logic, then they will perform better on a criterion test over the principle of mathematical induction than people who are not taught in terms of logic"Introduction.
Show less  Date Issued
 1972
 Identifier
 FSU_agg0249
 Format
 Thesis
 Title
 A comparison of verbal and nonverbal instruction in elementary school mathematics.
 Creator

Hollingsworth, Caroline Dean, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

This study assessed the relative effectiveness of verbal and nonverbal teaching methods in facilitating the learning of mathematics. The two treatments differed only in that nonverbal instruction did not permit oral communication or use of written words. Chalkboard instruction was characterized by complete silence in nonverbal classes. In verbal classes, new terminology was introduced by writing the terms on the board and using them thorughout the lesson. Four fourthgrade classes consisting...
Show moreThis study assessed the relative effectiveness of verbal and nonverbal teaching methods in facilitating the learning of mathematics. The two treatments differed only in that nonverbal instruction did not permit oral communication or use of written words. Chalkboard instruction was characterized by complete silence in nonverbal classes. In verbal classes, new terminology was introduced by writing the terms on the board and using them thorughout the lesson. Four fourthgrade classes consisting of 88 students in one school were randomly assigned to treatment groups so that two were taught nonverbally, and two by the conventional verbal method. Two teachers were assigned one class of each type. Treatment and teacher factors were crossed in a pretestposttest control group design. The demonstrated comparability of the two teaching methods not only points to nonverbal instruction as an alternate mode, but also seriously questions the effectiveness of conventional teacher talk in enhancing learning. Teachers with a creative bent should be encouraged to experiment with nonverbal instruction and design activities for all levels of development. The technique could be used effectively to break the routine of conventional instruction. The importance of nonverbal components should be stressed in methods courses for pre and inservice teachers. Techniques of nonverbal instruction should be practiced in student teaching practices.
Show less  Date Issued
 1973
 Identifier
 FSU_afa8702
 Format
 Thesis
 Title
 A study of the prediction of achievement in some topics in college freshman mathematics from measures of "structureofintellect" factors.
 Creator

Altman, Betty J., Nichols, Eugene Douglas, Florida State University
 Abstract/Description

For several reasons, Guilford's psychological theory, "The StructureofIntellect" (SI), seems a good candidate for relating to the learning of mathematics. The general purposes of this study were to identify SI factors which would be significantly related to achievement in a juniorcollege mathematics course for nonscience, nonmathematics majors and to determine whether semantic factors would be better predictors than symbolic for students classified as having high verbal ability. The two...
Show moreFor several reasons, Guilford's psychological theory, "The StructureofIntellect" (SI), seems a good candidate for relating to the learning of mathematics. The general purposes of this study were to identify SI factors which would be significantly related to achievement in a juniorcollege mathematics course for nonscience, nonmathematics majors and to determine whether semantic factors would be better predictors than symbolic for students classified as having high verbal ability. The two topics in the mathematics course which were selected for study were (1) numeration in other bases and (2) finite systems.
Show less  Date Issued
 1975
 Identifier
 FSU_abd5132
 Format
 Thesis
 Title
 The role of pictures in first grade children's perception of mathematical relationships.
 Creator

Campbell, Patricia F., Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"This study investigated whether there is a relationship between first grade children's ability to tell a story about a dynamic picture or a sequence of three dynamic pictures and their ability to describe the picture(s) by a number sequence. The artistic variables characterizing the pictures were controlled so as to provide information concerning which types of illustrations best facilitated interpretation of the pictures and perception of mathematical relationships. An 8 x 2 design allowed...
Show more"This study investigated whether there is a relationship between first grade children's ability to tell a story about a dynamic picture or a sequence of three dynamic pictures and their ability to describe the picture(s) by a number sequence. The artistic variables characterizing the pictures were controlled so as to provide information concerning which types of illustrations best facilitated interpretation of the pictures and perception of mathematical relationships. An 8 x 2 design allowed analysis of the effects of the form of the drawing, the number of pictures, and the response condition. Ninetysix first grade children were individually tested using an instrument designed by the investigator. Statistical analysis revealed that neither drawing style nor the number of pictures had a significant effect on either the level of assimilation within the stories, the perception of motion, or the number sentence responses. Analysis of the response condition revealed a significant difference favoring the force condition on number sentence responses. Also, initially viewing and interpreting sequences provided a learning experience to significantly effect the interpretation of single pictures"Abstract.
Show less  Date Issued
 1976
 Identifier
 FSU_abj0548
 Format
 Thesis
 Title
 Interactions between spatial and verbal abilities and two methods of presenting modulus seven arithmetic.
 Creator

Hussien, Gaber A, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"The present investigation was designed to study the effect of two instructional treatments on the achievement of students of different abilitiesVerbal and Spatial. This was achieved by studying the interaction between the two treatments and each of the verbal and the spatial abilities. The instructional treatments were Figural and Verbal programmed units designed to teach concepts related to modulus seven arithmetic. Subjects for the study were 90 students enrolled in the first year...
Show more"The present investigation was designed to study the effect of two instructional treatments on the achievement of students of different abilitiesVerbal and Spatial. This was achieved by studying the interaction between the two treatments and each of the verbal and the spatial abilities. The instructional treatments were Figural and Verbal programmed units designed to teach concepts related to modulus seven arithmetic. Subjects for the study were 90 students enrolled in the first year mathematics course at Elmansoura College of Education in Egypt for the academic year 19781979"Abstract.
Show less  Date Issued
 1979
 Identifier
 FSU_aby7218
 Format
 Thesis
 Title
 An investigation of the effect of instruction in the structure of problemsolving strategies on students' performance.
 Creator

Ghunaym, Ghunaym, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"The purpose of this study was to investigate the conjecture that instruction in the strategies of Pattern Discovery, Trial and Error, Working Backward, Contradiction, Substitution, and Use of Diagrams would result in the development of problemsolving ability and that students under this instruction are likely to exhibit better achievement than students who do not receive explicit instruction in problemsolving strategies"Introduction.
 Date Issued
 1985
 Identifier
 FSU_acr1501
 Format
 Thesis
 Title
 An exploratory study of the effectiveness of computer graphic and simulations in a computerstudent interactive environment in illustrating random sampling and the central limit theorem.
 Creator

Myers, Kitty Neel, Denmark, E. T., Florida State University
 Abstract/Description

"The purposes of this study were: (1) to investigate the effectiveness of the computerstudent interactive method in presenting statistical concepts and in instructing students in the applications of these concepts, and (2) to develop instruments that test for the understanding of these concepts and the mastery of these application skills"Abstract.
 Date Issued
 1990
 Identifier
 FSU_afs7567
 Format
 Thesis
 Title
 Using history in the teaching of mathematics.
 Creator

Awosanya, Ayokunle, Jakubowski, Elizabeth, Wills, Herbert, Florida State University
 Abstract/Description

The results reported here are the product of the research titled: Using history in the teaching of mathematics. The subjects are students in two classes of algebra II course at Florida State University High School 36 students makes and females whose ages are mostly 18 and a few 17 and 16 years old. Algebra II is a course that is usually taken by high school seniors in 12th grade and a few 11th or 10th grade students which explains why the ages of the students are mostly 18 and a few 17...
Show moreThe results reported here are the product of the research titled: Using history in the teaching of mathematics. The subjects are students in two classes of algebra II course at Florida State University High School 36 students makes and females whose ages are mostly 18 and a few 17 and 16 years old. Algebra II is a course that is usually taken by high school seniors in 12th grade and a few 11th or 10th grade students which explains why the ages of the students are mostly 18 and a few 17 and 16 years old. In this investigation, both quantitative study and qualitative study were employed. The quantitative study was the main study a teaching experiment using quasiexperimental methodology that involves two groups group 1 and group 2. Group 1 is the control group, where various algebraic/mathematical concepts, or topics were taught or explained to students with the necessary formulas. Group 2 was the experimental group in which the accounts of the historical origin of algebraic/mathematical concepts and the mathematicians (Lewis Carroll, Archimedes, Pythagoras, and Sophie Germain) who brought forward or created the concepts were used to augment pedagogical lessons and exercises used for this study as the main feature of pedagogy. The qualitative study augmented the main quantitative study; it was a followup interview for students to probe further an indepth rationale for the research theme, using history in the teaching of mathematics. The statistical analysis results indicated that there is a significant difference in the mean of score for the control group students and the mean of scores of the experimental group is greater than the mean on scores of student's performance in the control group; and the interview questions responses indeed corroborate the fact that the use of history in teaching mathematics does improve learning and understanding of algebraic/mathematical concepts.
Show less  Date Issued
 2001
 Identifier
 FSU_historic_akx6428
 Format
 Thesis
 Title
 Optimal Control of Continuous and Discontinuous Flow.
 Creator

Homescu, Cristian A., Navon, I. M., Pfeffer, R., Hussaini, M. Y., Erlebacher, G., Blumsack, S., Department of Mathematics, Florida State University
 Abstract/Description

Numerical and theoretical aspects of solving optimal control problems for a continuous flow (suppression of the Karman vortex street for a flow around a cylinder) and for a discontinuous flow (changing the location of discontinuities for the shocktube problem) are considered. The minimization algorithms require the gradient (or a subgradient) for the smooth (respectively non smooth) cost functional. The numerical value of the gradient (respectively a sub gradient) is obtained using the ad...
Show moreNumerical and theoretical aspects of solving optimal control problems for a continuous flow (suppression of the Karman vortex street for a flow around a cylinder) and for a discontinuous flow (changing the location of discontinuities for the shocktube problem) are considered. The minimization algorithms require the gradient (or a subgradient) for the smooth (respectively non smooth) cost functional. The numerical value of the gradient (respectively a sub gradient) is obtained using the adjoint method. The optimal solutions are verified using their physical interpretation. A very convincing argument for the validity of the numerical optimal solutions is obtained comparing the values corresponding to observed physical phenomena to the above mentioned numerical optimal controls. Sensitivity analysis of a discontinuous flow, namely for t he shocktube problem of gas dynamics, was also studied. Better results are obtained compared to the available literature, due to the use of adaptive mesh refinement.
Show less  Date Issued
 2002
 Identifier
 FSU_migr_etd4522
 Format
 Thesis
 Title
 Intersection Numbers of Divisors in Graph Varieties.
 Creator

Jones, Deborah, Aluffi, Paolo, Aldrovandi, Ettore, Hironaka., Eriko, Klassen, Eric, Reina, Laura, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations...
Show moreThis dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations explain a certain symmetry pattern we discovered.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd3426
 Format
 Thesis
 Title
 A Heuristic Method for a Rostering Problem with the Objective of Equal Accumulated Flying Time.
 Creator

Ye, Xugang, Blumsack, Steve, Bellenot, Steve, Braswell, Robert N., Department of Mathematics, Florida State University
 Abstract/Description

Crew costs are the second largest direct operating cost of airlines next to fuel costs. Therefore much research has been devoted to the planning and scheduling of crews over the last thirty years. The planning and scheduling of crews is a highly complex combinatorial problem that consists two independent phases. The first phase is the Crew Pairing Problem (CPP), which concerns finding a set of tasks with minimum cost while satisfying the service requirements. The second phase is the Crew...
Show moreCrew costs are the second largest direct operating cost of airlines next to fuel costs. Therefore much research has been devoted to the planning and scheduling of crews over the last thirty years. The planning and scheduling of crews is a highly complex combinatorial problem that consists two independent phases. The first phase is the Crew Pairing Problem (CPP), which concerns finding a set of tasks with minimum cost while satisfying the service requirements. The second phase is the Crew Rostering Problem (CRP), which concerns finding work assignment for crewmembers in a given period. In this thesis we focus on a Crew Rostering Problem, where a main pilot and a copilot perform a task. The model is a variance minimization problem with 01 variables and constraints associated with ensuring collective agreements, rules and guaranteeing the production of flights service. We choose a sequential constructive method (heuristic) to solve this difficult combinatorial problem since: (1), minimizing quadratic function of discrete variables makes linear methods difficult to use, a monthly schedule for one hundred pilots can generate tens of thousands variables and millions of constraints; (2), it is a NPhard problem, which means the CPU time of solution searching will grow exponentially as the instance dimension (the number of pilots and the number of tasks) increases. According to the characteristics of the model we propose, we do not find the global optimal solution; we find a satisfactory solution (or near optimal solution). The basic idea in our heuristic method is to decompose the assigning process into many subphases day by day. Then in dealing with minimizing the objective function, two heristic principals are employed. Meanwhile, in coping with the constraints, a weighted matching model and its algorithm will be used. In the numerical simulation, the comprehensive method is tested for its effectiveness. We show that our method can produce a solution whose objective value is below a satisfactory bound.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd0944
 Format
 Thesis
 Title
 Contour Modeling by Multiple Linear Regression of the Nineteen Piano Sonatas by Mozart.
 Creator

Beard, R. Daniel, Clendinning, Jane Piper, Song, KaiSheng, Mathes, James R., Spencer, Peter, College of Music, Florida State University
 Abstract/Description

Theories of musical contour can be described as the study of the change in one musical parameter as a function of another. In my dissertation, contour theories proposed by Robert Morris, Michael Friedmann, Elizabeth Marvin, Paul Laprade, Ian Quinn, Robert John Clifford, Larry Polansky and Richard Bassein are reviewed. In general, these authors approach changes in pitch as a function of time. A commonality between these theories was shown to be the use of a system of pitch level identification...
Show moreTheories of musical contour can be described as the study of the change in one musical parameter as a function of another. In my dissertation, contour theories proposed by Robert Morris, Michael Friedmann, Elizabeth Marvin, Paul Laprade, Ian Quinn, Robert John Clifford, Larry Polansky and Richard Bassein are reviewed. In general, these authors approach changes in pitch as a function of time. A commonality between these theories was shown to be the use of a system of pitch level identification based on the relative highness or lowness of the pitches, not based on actual pitch frequencies or pitch intervals in the melody. Additionally, these theories did not account for rhythmic or durational elements of the pitches as they are articulated in time. Music perception studies were cited that indicated that contour can play an important role in the recognition and memory of a melody, and that pitch interval and rhythmic components are vital elements in music understanding. Because these contour theories lacked the important musical elements of pitch and rhythm, an analytical method for the study of musical contour that incorporates both of these in its model of a melody is developed. This analytical method uses the mathematical technique of multiple linear regression to develop a model of the melody that can be graphed as representative of the contour of the actual melody. This method was used to analyze the first themes from the first movements of the nineteen piano sonatas composed by Mozart. Using regression modeling, the sonata melodies were categorized into two melody types: Type MD and Type LB. Analytical methods proposed by other theorists were then used to analyze selected melodies, and a comparison between the multiple linear regression model and these results was made.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd1173
 Format
 Thesis
 Title
 On the Geometry of Hurwitz Surfaces.
 Creator

Vogeler, Roger, Bowers, Philip L., Heil, Wolfgang H., Klassen, Eric P., Quine, John R., Srivastava, Anuj, Department of Mathematics, Florida State University
 Abstract/Description

A Riemann surface of genus g has at most 84(g − 1) automorphisms. A Hurwitz surface is one for which this maximum is attained; the corresponding group of automorphisms is called a Hurwitz group. By uniformization, the surface admits a hyperbolic structure wherein the automorphisms act by isometry. Such isometries descend from the (2,3,7) triangle group T acting on the universal cover H2. We develop a combinatorial approach which leads to a classification of the conjugacy classes of hyperbolic...
Show moreA Riemann surface of genus g has at most 84(g − 1) automorphisms. A Hurwitz surface is one for which this maximum is attained; the corresponding group of automorphisms is called a Hurwitz group. By uniformization, the surface admits a hyperbolic structure wherein the automorphisms act by isometry. Such isometries descend from the (2,3,7) triangle group T acting on the universal cover H2. We develop a combinatorial approach which leads to a classification of the conjugacy classes of hyperbolic elements of T, arranged by length. This allows us to study the closed geodesics of Hurwitz surfaces by performing calculations in the corresponding Hurwitz groups. We identify the systoles and other short curves on most of the Hurwitz surfaces of genus less than 10,000. We also determine which of these surfaces are chiral and which are amphichiral. In addition, we show that certain families of closed geodesics are simple on every Hurwitz surface.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd4544
 Format
 Thesis
 Title
 Envelopes, Duality, and Multipliers for Certain NonLocally Convex HardyLorentz Spaces.
 Creator

Lengfield, Marc, Oberlin, Daniel M., McKeague, Ian, Bellenot, Steven, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation is a study of the HardyLorentz spaces Hp,q for indices in the range 0 < p < 1, 0 < q ≤ ∞. More precisely, for p as indicated, we describe the dual spaces and Banach envelopes of the spaces Hp,q for finite values of q and do the same for H0p,∞, the closure of the polynomials in Hp,∞. In addition, we determine the sBanach envelopes for the spaces Hp,q in the cases 0 < q < p < s ≤ 1 and 0 < p < q ≤ s ≤ 1. As an application of our results we determine the multiplier spaces (Hp...
Show moreThis dissertation is a study of the HardyLorentz spaces Hp,q for indices in the range 0 < p < 1, 0 < q ≤ ∞. More precisely, for p as indicated, we describe the dual spaces and Banach envelopes of the spaces Hp,q for finite values of q and do the same for H0p,∞, the closure of the polynomials in Hp,∞. In addition, we determine the sBanach envelopes for the spaces Hp,q in the cases 0 < q < p < s ≤ 1 and 0 < p < q ≤ s ≤ 1. As an application of our results we determine the multiplier spaces (Hp,q, ℓs) for 0 < p < 1, 0 < q, s ≤ ∞.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3131
 Format
 Thesis
 Title
 Finite Abelian Group Actions on Orientable Circle Bundles over Surfaces.
 Creator

Ibrahim, Caroline Maher Boulis, Heil, Wolfgang, Hollander, Myles, Hironaka, Eriko, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

A finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that...
Show moreA finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that preserves the group action. This topological definition is translated to an algebraic definition as two Gactions are equivalent if and only if the associated surjections into G are equivalent via an automorphism of the fundamental group of M. For the manifolds M considered in this dissertation every automorphism of the fundamental group of M can be realized by a homeomorphism of M. Hence there is a onetoone correspondence between the topological and algebraic equivalence. The problem of classifying fixedpoint free finite abelian group actions on surfaces had been investigated by, among others, Nielsen, Smith and Zimmermann. Nielsen classifies cyclic actions on surfaces. He gives a list of automorphisms which he uses in his classification. Smith does the classification for special abelian groups. His approach is different from Nielsen's in the algebraic methods he uses. Zimmermann gives an algebraic solution to the classification of any finite abelian group action on closed surfaces. His technique is to get every surjective homomorphism from the fundamental group of the surface into G in normal form and then differentiate between the normal forms. In this dissertation we classify fixedpoint free finite abelian group actions on circle bundles. By results of Waldhausen every homeomorphism of M is isotopic to a fiber preserving homeomorphism; that is, it preserves the S1 factor of the bundle. This corresponds to the algebraic condition that any automorphism on the fundamental group of M preserves the center of the group. We use the same approach as that of Nielsen on surfaces. We give algorithms to bring every surjective homomorphism from the fundamental group of the bundle into the group G to normal form. From there we differentiate between the normal forms based on Nielsen's results. The results obtained are for circle bundles over surfaces of genus g greater than or equal to 2. A complete classification is given for the case that the circle bundle is a product bundle and G is a finite abelian group. We also obtain a complete classification of cyclic group actions and finite abelian group actions on circle bundles that are not product bundles.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3887
 Format
 Thesis
 Title
 Steady Dynamics in Shearing Flows of Nematic Liquid Crystalline Polymers.
 Creator

Liu, Fangyu, Wang, Qi, Sussman, Mark, Song, Kaisheng, Department of Mathematics, Florida State University
 Abstract/Description

The biaxiality of the steady state solutions and their stability to inplane disturbances in shearing flows of nematic liquid crystalline polymers are studied by using simplified Wang (2002) model. We obtain all the steady states of Wang model exhibit biaxial symmetry in which two directors are confined to the shearing plane and analysis their stability with respect to inplane disturbances at isolated Debra numbers and polymer concentration values.
 Date Issued
 2004
 Identifier
 FSU_migr_etd1190
 Format
 Thesis
 Title
 A Stock Market AgentBased Model Using Evolutionary Game Theory and Quantum Mechanical Formalism.
 Creator

Montin, Benoit S., Nolder, Craig A., Huﬀer, Fred W., Case, Bettye Anne, Beaumont, Paul M., Kercheval, Alec N., Sumners, DeWitt L., Department of Mathematics, Florida State...
Show moreMontin, Benoit S., Nolder, Craig A., Huﬀer, Fred W., Case, Bettye Anne, Beaumont, Paul M., Kercheval, Alec N., Sumners, DeWitt L., Department of Mathematics, Florida State University
Show less  Abstract/Description

The financial market is modelled as a complex selforganizing system. Three economic agents interact in a simplified economy and seek the maximization of their wealth. Replicator dynamics are used as a myopic behavioral rule to describe how agents learn and benefit from their experiences. Stock price fluctuations result from interactions between economic agents, budget constraints and conservation laws. Time is discrete. Invariant distributions over the state space, that is to say probability...
Show moreThe financial market is modelled as a complex selforganizing system. Three economic agents interact in a simplified economy and seek the maximization of their wealth. Replicator dynamics are used as a myopic behavioral rule to describe how agents learn and benefit from their experiences. Stock price fluctuations result from interactions between economic agents, budget constraints and conservation laws. Time is discrete. Invariant distributions over the state space, that is to say probability measures that remain unchanged by the oneperiod transition rule, form stochastic equilibria for our composite system. When agents make mistakes, there is a unique stochastic steady state which reflects the average and limit behavior. Convergence of the iterates occurs at a geometric rate in the total variation norm. Interestingly, when the probability of making a mistake tends to zero, the invariant distribution converges weakly to a stochastic equilibrium for the model without mistakes. Most agentbased computational economies heavily rely on simulations. Having adopted a simple representation of financial markets, we have been able to prove the above theoretical results and gain intuition on complexity economics. The impact of simple monetary policies on the limit stock price distribution, such as a decrease of the riskfree rate of interest, has been analyzed. Of interest as well, the limit stock log return distribution presents realworld features (skewed and leptokurtic) that more traditional models usually fail to explain or consider. Our artificial market is incomplete. The bid and ask prices of a vanilla Call option have been computed to illustrate option pricing in our setting.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd2331
 Format
 Thesis
 Title
 All Speed MultiPhase Flow Solvers.
 Creator

Kadioglu, Samet Y., Sussman, Mark, Telotte, John, Hussaini, Yousuﬀ, Wang, Qi, Erlebacher, Gordon, Department of Mathematics, Florida State University
 Abstract/Description

A new second order primitive preconditioner technique (an all speed method) for solving all speed single/multiphase flow is presented. With this technique, one can compute both compressible and incompressible flows with Machuniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach numbers). The new primitive preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks, providing highly resolved shock solutions together...
Show moreA new second order primitive preconditioner technique (an all speed method) for solving all speed single/multiphase flow is presented. With this technique, one can compute both compressible and incompressible flows with Machuniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach numbers). The new primitive preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks, providing highly resolved shock solutions together with correct shock speeds. In addition, the new technique performs very well at the zero Mach limit. In the case of multiphase flow, the new primitive preconditioner technique enables one to accurately treat phase boundaries in which there is a large impedance mismatch. When solving multidimensional all speed multiphase flows, we introduce adaptive solution techniques which exploit the advantages of Machuniform methods. We compute a variety of problems from low (low speed) to high Mach number (high speed) flows including multiphase flow tests, i.e, computing the growth and collapse of adiabatic bubbles for study of underwater explosions
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3391
 Format
 Thesis
 Title
 Calibration of Multivariate Generalized Hyperbolic Distributions Using the EM Algorithm, with Applications in Risk Management, Portfolio Optimization and Portfolio Credit Risk.
 Creator

Hu, Wenbo, Kercheval, Alec, Huﬀer, Fred, Case, Bettye, Nichols, Warren, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

The distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical...
Show moreThe distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical difficulty of calibrating the distributional parameters to the data. In this dissertation we describe a way to stably calibrate GH distributions for a wider range of parameters than has previously been reported. In particular, we develop a version of the EM algorithm for calibrating GH distributions. This is a modification of methods proposed in McNeil, Frey, and Embrechts (2005), and generalizes the algorithm of Protassov (2004). Our algorithm extends the stability of the calibration procedure to a wide range of parameters, now including parameter values that maximize loglikelihood for our real market data sets. This allows for the first time certain GH distributions to be used in modeling contexts when previously they have been numerically intractable. Our algorithm enables us to make new uses of GH distributions in three financial applications. First, we forecast univariate ValueatRisk (VaR) for stock index returns, and we show in outofsample backtesting that the GH distributions outperform the Gaussian distribution. Second, we calculate an efficient frontier for equity portfolio optimization under the skewedt distribution and using Expected Shortfall as the risk measure. Here, we show that the Gaussian efficient frontier is actually unreachable if returns are skewed t distributed. Third, we build an intensitybased model to price Basket Credit Default Swaps by calibrating the skewed t distribution directly, without the need to separately calibrate xi the skewed t copula. To our knowledge this is the first use of the skewed t distribution in portfolio optimization and in portfolio credit risk.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3694
 Format
 Thesis
 Title
 Singleand MultipleObjective Stochastic Programming Models with Applications to Aerodynamics.
 Creator

Croicu, AnaMaria, Hussaini, M. Yousuﬀ, Srivastava, Anuj, Kopriva, David, Wang, Qi, Department of Mathematics, Florida State University
 Abstract/Description

Deterministic design assumes that there is no uncertainty in the modeling parameters, and as a consequence, there is no variability in the simulation outputs. Therefore, deterministic optimal designs that are obtained without taking into account uncertainty are usually unreliable. This is the case with transonic shape optimization, where the randomness in the cruise Mach number might have significant impact on the optimal geometric design. In this context, a stochastic search turns out to be...
Show moreDeterministic design assumes that there is no uncertainty in the modeling parameters, and as a consequence, there is no variability in the simulation outputs. Therefore, deterministic optimal designs that are obtained without taking into account uncertainty are usually unreliable. This is the case with transonic shape optimization, where the randomness in the cruise Mach number might have significant impact on the optimal geometric design. In this context, a stochastic search turns out to be more appropriate. Approaches to stochastic optimization have followed a variety of modeling philosophies, but little has been done to systematically compare different models. The goal of this thesis is to present a comparison between two stochastic optimization algorithms, with the emphasis on applications, especially on the airfoil shape optimization. Singleobjective and multiobjective optimization programs are analyzed as well. The relationship between the expected minimum value (EMV) criterion and the minimum expected value (MEV) criterion is explored, and it is shown that, under favorable conditions, a better optimal point could be obtained via the EMV approach. Unfortunately, the advantages of using the EMV approach are far outweighed by the prohibitive exorbitant computational cost.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3027
 Format
 Thesis
 Title
 Rheology and Mesoscale Morphology of Flows of Chlesteric and Nematic Liquid Crystal Polymers.
 Creator

Cui, Zhenlu, Wang, Qi, Liu, Guosheng, Magnan, Jerry F., Sussman, Mark, Tam, Christopher, Department of Mathematics, Florida State University
 Abstract/Description

Cholesteric liquid crystals(CLC) are mesophases, where the average direction of molecular orientation exhibits a chiral (twisted) pattern along its normal direction. In the past, the rheological and flow properties of CLC have been studied scarcely. This is due to the natural tendency of a cholesteric to favor its characteric, twisted configuration, which naturally leads to more complex arrangements of the optic axis than in pure nematics and complicated spatial structures. In this...
Show moreCholesteric liquid crystals(CLC) are mesophases, where the average direction of molecular orientation exhibits a chiral (twisted) pattern along its normal direction. In the past, the rheological and flow properties of CLC have been studied scarcely. This is due to the natural tendency of a cholesteric to favor its characteric, twisted configuration, which naturally leads to more complex arrangements of the optic axis than in pure nematics and complicated spatial structures. In this dissertation, we address the issues related to rheology and flow induced structures in CLC and nematic polymers, with emphasis on the role of the anisotropic elasticities. In the first part of this dissertation, we study the permeation flow problem using a mesoscopic theory obtained from the kinetic theory for Cholesteric liquid crystal polymers and resolve the inconsistency issue in the literature. Then we give a systematic study on steady structures and transient behavior in flows of nematic polymers. In the second part of this dissertation, we develop a hydrodynamic theory for flows of CLCPs following the continuum mechanics formulation of McMillan's second order tensor theory for liquid crystals and study phase transition in chiral nematic liquid crystals as well as the rheological hebaviors and the flow properties of CLCPs.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd2952
 Format
 Thesis
 Title
 I. A Modified ƙƐ Turbulence Model for High Speed Hets at Elevated Temperatures. II. Modeling and a Computational Study of Spliced Acoustic Liners.
 Creator

Ganesan, Anand, Tam, Christopher K. W., Nh, HonKie, Hunter, Christopher, Navon, Ionel Michael, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

A modification to the kepsilon model aimed to extend its applicability to the computation of the mean flow and noise of highspeed hot jets is proposed. The motivation of the proposal arises from the observation that there is a large density induced increase in the growth rate of spatial instabilities of a mixing layer if the lighter fluid moves faster. This consideration leads to the incorporation of a density gradient related contribution to the turbulent eddy viscosity of the kepsilon...
Show moreA modification to the kepsilon model aimed to extend its applicability to the computation of the mean flow and noise of highspeed hot jets is proposed. The motivation of the proposal arises from the observation that there is a large density induced increase in the growth rate of spatial instabilities of a mixing layer if the lighter fluid moves faster. This consideration leads to the incorporation of a density gradient related contribution to the turbulent eddy viscosity of the kepsilon model. Computed jet mean flow profiles and centerline velocity distributions at elevated temperatures of highspeed jets are found to be in better agreement with experimental measurements if density modification is included. Noise predictions including density effect are also found to be in better agreement with microphone measurements. The good agreements offer strong support to the validity and usefulness of the proposed density correction formula. A timedomain computational methodology has been deveoped to study the propagation and acoustic scattering of fan tones by spliced liners. The front portion of the engine is modelled as a duct. Significant acoustic scattering is observed for a frequency pretty close to cutoff. In this case, total scattered energy was found to be more than the energy in the incident mode. The spliced liners, in such conditions, are found to be less effective than the uniform liners. The performance of the liner was found to be dependent on the frequency. The results of the simulations agree qualitiatively well with the available experimental and theoretical work.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd4368
 Format
 Thesis
 Title
 Applications of Representation Theory and HigherOrder Perturbation Theory in NMR.
 Creator

Srinivasan, Parthasarathy, Quine, John R., Gan, Zhehong, Chapman, Michael S., Bowers, Philip, Sumners, DeWitt, Department of Mathematics, Florida State University
 Abstract/Description

Solid State Nuclear Magnetic Resonance (NMR) is perhaps the only spectroscopic technique that allows experimentalists to manipulate the spin systems they are interested in. Of particular interest are nuclei with spins greater than 1/2, or quadrupolar nuclei, as they constitute over 70% of the magnetically active spins. Two of the important mathematical tools used in the theory of studying NMR are representation theory together with perturbation theory. We will use both these tools to describe...
Show moreSolid State Nuclear Magnetic Resonance (NMR) is perhaps the only spectroscopic technique that allows experimentalists to manipulate the spin systems they are interested in. Of particular interest are nuclei with spins greater than 1/2, or quadrupolar nuclei, as they constitute over 70% of the magnetically active spins. Two of the important mathematical tools used in the theory of studying NMR are representation theory together with perturbation theory. We will use both these tools to describe the underlying mathematical theory for quadrupolar nuclei. The theory shows that for nonsymmetric satellite transitions in halfinteger quadrupolar nuclei, perturbation effects up to thirdorder feature in the NMR spectra. We will also use irreducible representations to analyze experiments conducted on various spin systems and discuss ways to design new ones. Another topic that will also be explored is the theory of rotary resonance in halfinteger quadrupolar nuclei. This theory explains why techniques like FASTER (FAster Spinning gives Transfer Enhancement at Rotary resonance) improve the efficiency of symmetric multiple quantum experiments.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd1600
 Format
 Thesis
 Title
 The Fractal Nature of Lightning: An Investigation of the Fractal Relationship of the Structure of Lightning to Terrain.
 Creator

GrahamJones, Brian Clay, Hunter, Christopher, Elsner, James B., Bellenot, Steve, Department of Mathematics, Florida State University
 Abstract/Description

This study focuses on the relationship between the structure of lightning and how it may or may not be related to the topography below it.
 Date Issued
 2006
 Identifier
 FSU_migr_etd4055
 Format
 Thesis
 Title
 Combinatorial Type Problems for Triangulation Graphs.
 Creator

Wood, William E., Bowers, Philip, Hawkes, Lois, Bellenot, Steve, Klassen, Eric, Nolder, Craig, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The main result in this thesis bounds the combinatorial modulus of a ring in a triangulation graph in terms of the modulus of a related ring. The bounds depend only on how the rings are related and not on the rings themselves. This may be used to solve the combinatorial type problem in a variety of situation, most significant in graphs with unbounded degree. Other results regarding the type problem are presented along with several examples illustrating the limits of the results.
 Date Issued
 2006
 Identifier
 FSU_migr_etd0794
 Format
 Thesis
 Title
 Predegree Polynomials of Plane Configurations in Projective Space.
 Creator

Tzigantchev, Dimitre G. (Dimitre Gueorguiev), Aluﬃ, Paolo, Reina, Laura, Aldrovandi, Ettore, Klassen, Eric, Seppälä, Mika, Department of Mathematics, Florida State University
 Abstract/Description

We work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the image of a rational map from P^15 to P^N. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general P^j , j being the...
Show moreWe work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the image of a rational map from P^15 to P^N. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general P^j , j being the dimension of the orbit. We find the predegrees and other invariants for all surfaces supported on unions of planes. The information is encoded in the socalled adjusted predegree polynomials, which possess nice multiplicative properties allowing us to easily compute the predegree (polynomials) of various special plane configurations. The predegree has both a combinatorial and geometric significance. The results obtained in this thesis would be a necessary step in the solution of the problem of computing predegrees for all surfaces.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd1747
 Format
 Thesis
 Title
 Option Pricing with Selfsimilar Additive Processes.
 Creator

Galloway, Mack L. (Mack Laws), Nolder, Craig, Huﬀer, Fred, Beaumont, Paul, Case, Bettye Anne, Quine, John R., Department of Mathematics, Florida State University
 Abstract/Description

The use of timeinhomogeneous additive models in option pricing has gained attention in recent years due to their potential to adequately price options across both strike and maturity with relatively few parameters. In this thesis two such classes of models based on the selfsimilar additive processes of Sato are developed. One class of models consists of the riskneutral exponentials of a selfsimilar additive process, while the other consists of the riskneutral exponentials of a Brownian...
Show moreThe use of timeinhomogeneous additive models in option pricing has gained attention in recent years due to their potential to adequately price options across both strike and maturity with relatively few parameters. In this thesis two such classes of models based on the selfsimilar additive processes of Sato are developed. One class of models consists of the riskneutral exponentials of a selfsimilar additive process, while the other consists of the riskneutral exponentials of a Brownian motion timechanged by an independent, increasing, selfsimilar additive process. Examples from each class are constructed in which the time one distributions are Variance Gamma or Normal Inverse Gaussian distributed. Pricing errors are assessed for the case of Standard and Poor's 500 index options from the year 2005. Both sets of timeinhomogeneous additive models show dramatic improvement in pricing error over their associated Lévy processes. Furthermore, with regard to the average of the pricing errors over the quote dates studied, the selfsimilar Normal Inverse Gaussian model yields a mean pricing error significantly less than that implied by the bidask spreads of the options, and also significantly less than that given by its associated, less parsimonious, Lévy stochastic volatility model.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd4372
 Format
 Thesis
 Title
 Analysis of Orientational Restraints in SolidState Nuclear Magnetic Resonance with Applications to Protein Structure Determination.
 Creator

Achuthan, Srisairam, Quine, John R., Cross, Timothy A., Sumners, DeWitt, Bertram, Richard, Department of Mathematics, Florida State University
 Abstract/Description

Of late, pathbreaking advances are taking place and flourishing in the field of solidstate Nuclear Magnetic Resonance (ssNMR)spectroscopy. One of the major applications of ssNMR techniques is to high resolution threedimensional structures of biological molecules like the membrane proteins. An explicit example of this is PISEMA (Polarization Inversion Spin Exchange at Magic Angle). This dissertation studies and analyzes the use of the orientational restraints in general, and particularly...
Show moreOf late, pathbreaking advances are taking place and flourishing in the field of solidstate Nuclear Magnetic Resonance (ssNMR)spectroscopy. One of the major applications of ssNMR techniques is to high resolution threedimensional structures of biological molecules like the membrane proteins. An explicit example of this is PISEMA (Polarization Inversion Spin Exchange at Magic Angle). This dissertation studies and analyzes the use of the orientational restraints in general, and particularly the restraints measured through PISEMA. Here, we have applied our understanding of orientational restraints to briefly investigate the structure of Amantadine bound M2TMD, a membrane protein in Influenza A Virus. We model the protein backbone structure as a discrete curve in space with atoms represented by vertices and covalent bonds connecting them as the edges. The oriented structure of this curve with respect to an external vector is emphasized. The map from the surface of the unit sphere to the PISEMA frequency plane is examined in detail. The image is a powder pattern in the frequency plane. A discussion of the resulting image is provided. Solutions to PISEMA equations lead to multiple orientations for the magnetic field vector for a given point in the frequency plane. These are duly captured by sign degeneracies for the vector coordinates. The intensity of NMR powder patterns is formulated in terms of a probability density function for 1d spectra and a joint probability density function for the 2d spectra. The intensity analysis for 2d spectra is found to be rather helpful in addressing the robustness of the PISEMA data. To build protein structures by gluing together diplanes, certain necessary conditions have to be met. We formulate these as continuity conditions to be realized for diplanes. The number of oriented protein structures has been enumerated in the degeneracy framework for diplanes. Torsion angles are expressed via sign degeneracies. For aligned protein samples, the PISA wheel approach to modeling the protein structure is adopted. Finally, an atomic model of the monomer structure of M2TMD with Amantadine has been elucidated based on PISEMA orientational restraints. This is a joint work with Jun Hu and Tom Asbury. The PISEMA data was collected by Jun Hu and the molecular modeling was performed by Tom Asbury.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd0109
 Format
 Thesis
 Title
 Deterministic and Stochastic Aspects of Data Assimilation.
 Creator

Akella, Santharam, Navon, Ionel Michael, O'Brien, James J., Erlebacher, Gordon, Wang, Qi, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

The principles of optimal control of distributed parameter systems are used to derive a powerful class of numerical methods for solutions of inverse problems, called data assimilation (DA) methods. Using these DA methods one can efficiently estimate the state of a system and its evolution. This information is very crucial for achieving more accurate long term forecasts of complex systems, for instance, the atmosphere. DA methods achieve their goal of optimal estimation via combination of all...
Show moreThe principles of optimal control of distributed parameter systems are used to derive a powerful class of numerical methods for solutions of inverse problems, called data assimilation (DA) methods. Using these DA methods one can efficiently estimate the state of a system and its evolution. This information is very crucial for achieving more accurate long term forecasts of complex systems, for instance, the atmosphere. DA methods achieve their goal of optimal estimation via combination of all available information in the form of measurements of the state of the system and a dynamical model which describes the evolution of the system. In this dissertation work, we study the impact of new nonlinear numerical models on DA. High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. The impact of high resolution advection schemes in the framework of inverse problem solution/ DA has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on DA is the subject of this work. In 1D we use a nonlinear viscous Burgers equation and in 2D a global nonlinear shallow water model has been used. The results obtained show that using the various advection schemes consistently improves variational data assimilation (VDA) in the strong constraint form, which does not include model error. However, the cost functional included efficient and physically meaningful construction of the background cost functional term, J_b, using balance and diffusion equation based correlation operators. This was then followed by an indepth study of various approaches to model the systematic component of model error in the framework of a weak constraint VDA. Three simple forms, decreasing, invariant, and exponentially increasing in time forms of evolution of model error were tested. The inclusion of model error provides a substantial reduction in forecasting errors, in particular the exponentially increasing form in conjunction with the piecewise parabolic high resolution advection scheme was found to provide the best results. Results obtained in this work can be used to formulate sophisticated forms of model errors, and could lead to implementation of new VDA methods using numerical weather prediction models which involve high resolution advection schemes such as the van Leer slope limiters and the PPM.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd0145
 Format
 Thesis
 Title
 Analysis and Approximation of a TwoBand GinzburgLandau Model of Superconductivity.
 Creator

Chan, WanKan, Gunzburger, Max, Peterson, Janet, Manousakis, Efstratios, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

In 2001, the discovery of the intermetallic compound superconductor MgB2 having a critical temperature of 39K stirred up great interest in using a generalization of the GinzburgLandau model, namely the twoband timedependent GinzburgLandau (2BTDGL) equations, to model the phenomena of twoband superconductivity. In this work, various mathematical and numerical aspects of the twodimensional, isothermal, isotropic 2BTDGL equations in the presence of a timedependent applied magnetic field...
Show moreIn 2001, the discovery of the intermetallic compound superconductor MgB2 having a critical temperature of 39K stirred up great interest in using a generalization of the GinzburgLandau model, namely the twoband timedependent GinzburgLandau (2BTDGL) equations, to model the phenomena of twoband superconductivity. In this work, various mathematical and numerical aspects of the twodimensional, isothermal, isotropic 2BTDGL equations in the presence of a timedependent applied magnetic field and a timedependent applied current are investigated. A new gauge is proposed to facilitate the inclusion of a timedependent current into the model. There are three parts in this work. First, the 2BTDGL model which includes a timedependent applied current is derived. Then, assuming sufficient smoothness of the boundary of the domain, the applied magnetic field, and the applied current, the global existence, uniqueness and boundedness of weak solutions of the 2BTDGL equations are proved. Second, the existence, uniqueness, and stability of finite element approximations of the solutions are shown and error estimates are derived. Third, numerical experiments are presented and compared to some known results which are related to MgB2 or general twoband superconductivity. Some novel behaviors are also identified.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3923
 Format
 Thesis
 Title
 Biomedical Applications of Shape Descriptors.
 Creator

Celestino, Christian Edgar Laing, Sumners, De Witt, Greenbaum, Nancy, Mio, Washington, Hurdal, Monica, Department of Mathematics, Florida State University
 Abstract/Description

Given an edgeoriented polygonal graph in R3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of pathconnected open regions on the unit sphere. Within each open region, the directional writhe is constant. We developed formulas for the writhe of polygons on Bravais lattices and a few crystallographic groups, and discuss...
Show moreGiven an edgeoriented polygonal graph in R3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of pathconnected open regions on the unit sphere. Within each open region, the directional writhe is constant. We developed formulas for the writhe of polygons on Bravais lattices and a few crystallographic groups, and discuss applications to ring polymers. In addition, we obtained a closed formula for the writhe for graphs which extends the formula for the writhe of a polygon in R3, including the important special case of writhe of embedded open arcs. Additionally, we have developed shape descriptors based on a family of geometric measures for the purpose of classification and identification of shape differences for graphs. These shape descriptors involve combinations of writhe and average crossing numbers of curves, as well as total curvature, ropelength and thickness. We have applied these shape descriptors to RNA tertiary structures and families of sulcal curves from human brain surfaces. Preliminary results give an automatic method to distinguish RNA motifs. Clear differentiation among tRNA and/or ribozymes, and a distinction among mesophilic and thermophilic tRNA is shown. In addition, we notice a direct correlation between the length of an RNA backbone and its mean average crossing number which is described accurately by a power function. As a neuroscience application, human brain surfaces were extracted from MRI scans of human brains. In our preliminary results, an automatic differentiation between sulcal paths from the left or right hemispheres, an age differentiation and a malefemale classification were achieved.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3314
 Format
 Thesis
 Title
 DNA Knotting: Occurrences, Consequences & Resolution.
 Creator

Mann, Jennifer Katherine, Sumners, De Witt L., Zechiedrich, E. Lynn, Greenbaum, Nancy L., Heil, Wolfgang, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation applies knot theory, DNA topology, linear algebra, statistics, probability theory and statistical mechanics to address questions about knotted, doublestranded DNA. The three main investigations are the cellular effects of knotting, the biophysics of knotting/unknotting and the unknotting mechanism of human topoisomerase IIá. The cellular effects of knotting were done in collaboration with Rick Deibler. The statistical mechanics were done in collaboration with Zhirong Liu...
Show moreThis dissertation applies knot theory, DNA topology, linear algebra, statistics, probability theory and statistical mechanics to address questions about knotted, doublestranded DNA. The three main investigations are the cellular effects of knotting, the biophysics of knotting/unknotting and the unknotting mechanism of human topoisomerase IIá. The cellular effects of knotting were done in collaboration with Rick Deibler. The statistical mechanics were done in collaboration with Zhirong Liu and Hue Sun Chan. Cellular DNA knotting is driven by DNA compaction, topoisomerization, replication, supercoilingpromoted strand collision, and DNA selfinteractions resulting from transposition, sitespecific recombination, and transcription (Spengler, Stasiak, and Cozzarelli 1985; Heichman, Moskowitz, and Johnson 1991; Wasserman and Cozzarelli 1991; Sogo, Stasiak, MartinezRobles et al. 1999). Type II topoisomerases are ubiquitous, essential enzymes that interconvert DNA topoisomers to resolve knots. These enzymes pass one DNA helix through another by creating an enzymebridged transient break. Explicitly how type II topoisomerases recognize their substrate and decide where to unknot DNA is unknown. What are the biological consequences of unresolved cellular DNA knotting? We investigated the physiological consequences of the wellaccepted propensity of cellular DNA to collide and react with itself by analyzing the effects of plasmid recombination and knotting in E. coli using a sitespecific recombination system. Fluctuation assays were performed to determine mutation rates of the strains used in these experiments (Rosche and Foster 2000). Our results show that DNA knotting: (i) promotes replicon loss by blocking DNA replication, (ii) blocks gene transcription, (iii) increases antibiotic sensitivity and (iv) promotes genetic rearrangements at a rate which is four orders of magnitude greater than of an unknotted plasmid. If unresolved, DNA knots can be lethal and may help drive genetic evolution. The faster and more efficiently type II topoisomerase unknots, the less chance for these disastrous consequences. How do type II topoisomerases unknot, rather than knot? If type II topoisomerases act randomly on juxtapositions of two DNA helices, knots are produced with probability depending on the length of the circular DNA substrate. For example, random strand passage is equivalent to random cyclization of linear substrate, and random cyclization of 10.5 kb substrate produces about 3% DNA knots, mostly trefoils (Rybenkov, Cozzarelli, and Vologodskii 1993; Shaw and Wang 1993). However, experimental data show that type II topoisomerases unknot at a level up to 90fold the level achieved by steadystate random DNA strand passage (Rybenkov, Ullsperger, and Vologodskii et al. 1997). Various models have been suggested to explain these results and all of them assume that the enzyme directs the process. In contrast, our laboratory proposed (Buck and Zechiedrich 2004) that type II topoisomerases recognize the curvature of the two DNA helices within a juxtaposition and the resulting angle between the helices. Furthermore, the values of curvature and angle lie within their respective bounds, which are characteristic of DNA knots. Thus, our model uniquely proposes unknotting is directed by the DNA and not the protein. We used statistical mechanics to test this hypothesis. Using a lattice polymer model, we generated conformations from preexisting juxtaposition geometries and studied the resulting knot types. First we determined the statistical relationship between the local geometry of a juxtaposition of two chain segments and whether the loop is knotted globally. We calculated the HOMFLY (Freyd, Yetter, and Hoste et al. 1985) polynomial of each conformation to identify knot types. We found that hooked juxtapositions are far more likely to generate knots than free juxtapositions. Next we studied the transitions between initial and final knot/unknot states that resulted from a type II topoisomeraselike segment passage at the juxtaposition. Selective segment passages at free juxtapositions tended to increase knot probability. In contrast, segment passages at hooked juxtapositions caused more transitions from knotted to unknot states than vice versa, resulting in a steadystate knot probability much less than that at topological equilibrium. In agreement with experimental type II topoisomerase results, the tendency of a segment passage at a given juxtaposition to unknot is strongly correlated with the tendency of that segment passage to decatenate. These quantitative findings show that there exists discriminatory topological information in local juxtaposition geometries that could be utilized by the enzyme to unknot rather than knot. This contrasts with prior thought that the enzyme itself directs unknotting and strengthens the hypothesis proposed by our group that type II topoisomerases act on hooked rather than free juxtapositions. Will a type II topoisomerase resolve a DNA twist knot in one cycle of action? The group of knots known as twist knots is intriguing from both knot theoretical and biochemical perspectives. A twist knot consists of an interwound region with any number of crossings and a clasp with two crossings. By reversing one of the crossings in the clasp the twist knot is converted to the unknot. However, a crossing change in the interwound region produces a twist knot with two less nodes. Naturally occurring knots in cells are twist knots. The unknotting number, the minimal number of crossing reversals required to convert a knot to the unknot, is equal to one for any twist knot. Each crossing reversal performed by a type II topoisomerase requires energy. Within the cell, DNA knots might be pulled tight by forces such as those which accompany transcription, replication and segregation, thus increasing the likelihood of DNA damage. Therefore, it would be advantageous for type II topoisomerases to act on a crossing in the clasp region of a DNA twist knot, thus, resolving the DNA knot in a single step. The mathematical unknotting number corresponds to the smallest number of topoisomerase strand passage events needed to untie a DNA knot. In order to study unknotting of DNA knots by a type II topoisomerase, I used sitespecific recombination systems and a benchtop fermentor to isolate large quantities of knotted DNA. My data show that purified five and sevennoded twist knots are converted to the unknot by human topoisomerase IIá with no appearance of either trefoils or fivenoded twist knots which are possible intermediates if the enzyme acted on one of the interwound nodes. Consequently, these data suggest that type II topoisomerase may preferentially act upon the clasp region of a twist knot. We have uniquely combined biology, chemistry, physics and mathematics to gain insight into the mechanism of type II topoisomerases, which are an important class of drug targets. Our results suggest that DNA knotting alters DNA structure in a way that may drive type II topoisomerase resolution of DNA knots. Ultimately, the knowledge gained about type II topoisomerases and their unknotting mechanism may lead to the development of new drugs and treatments of human infectious diseases and cancer.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd2754
 Format
 Thesis
 Title
 Asset Pricing in a Lucas Framework with Boundedly Rational, Heterogeneous Agents.
 Creator

Culham, Andrew J. (Andrew James), Beaumont, Paul M., Kercheval, Alec N., Schlagenhauf, Don, Goncharov, Yevgeny, Kopriva, David, Department of Mathematics, Florida State University
 Abstract/Description

The standard dynamic general equilibrium model of financial markets does a poor job of explaining the empirical facts observed in real market data. The common assumptions of homogeneous investors and rational expectations equilibrium are thought to be major factors leading to this poor performance. In an attempt to relax these assumptions, the literature has seen the emergence of agentbased computational models where artificial economies are populated with agents who trade in stylized asset...
Show moreThe standard dynamic general equilibrium model of financial markets does a poor job of explaining the empirical facts observed in real market data. The common assumptions of homogeneous investors and rational expectations equilibrium are thought to be major factors leading to this poor performance. In an attempt to relax these assumptions, the literature has seen the emergence of agentbased computational models where artificial economies are populated with agents who trade in stylized asset markets. Although they offer a great deal of flexibility, the theoretical community has often criticized these agentbased models because the agents are too limited in their analytical abilities. In this work, we create an artificial market with a single risky asset and populate it with fully optimizing, forward looking, infinitely lived, heterogeneous agents. We restrict the state space of our agents by not allowing them to observe the aggregate distribution of wealth so they are required to compute their conditional demand functions while simultaneously learning the equations of motion for the aggregate state variables. We develop an efficient and flexible model code that can be used to explore a wide number of asset pricing questions while remaining consistent with conventional asset pricing theory. We validate our model and code against known analytical solutions as well as against a new analytical result for agents with differing discount rates. Our simulation results for general cases without known analytical solutions show that, in general, agents' asset holdings converge to a steadystate distribution and the agents are able to learn the equilibrium prices despite the restricted state space. Further work will be necessary to determine whether the exceptional cases have some fundamental theoretical explanation or can be attributed to numerical issues. We conjecture that convergence to the equilibrium is global and that the marketclearing price acts to guide the agents' forecasts toward that equilibrium.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd2948
 Format
 Thesis
 Title
 Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data.
 Creator

Webster, Clayton G. (Clayton Garrett), Gunzburger, Max D., Gallivan, Kyle, Peterson, Janet, Tempone, Raul, Department of Mathematics, Florida State University
 Abstract/Description

The objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawCurtis or Gaussian abscissas. Even in the presence of...
Show moreThe objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawCurtis or Gaussian abscissas. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. The full tensor product spaces suffer from the curse of dimensionality since the dimension of the approximating space grows exponentially in the number of random variables. When this number is moderately large, we combine the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh equally in the solution; the latter approach is ideal when solving highly anisotropic problems depending on a relatively small number of random variables. We also include a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each problem. These procedures are very effective for the problems under study. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and compare this approach with several others, including the standard Monte Carlo. For moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very efficient and superior to all examined methods. Due to the high cost of effecting each realization of the PDE this work also proposes the use of reducedorder models (ROMs) that assist in minimizing the cost of determining accurate statistical information about outputs from ensembles of realizations. We explore the use of ROMs, that greatly reduce the cost of determining approximate solutions, for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decompositionbased ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd1223
 Format
 Thesis
 Title
 Geometric and Computational Generation, Correction, and Simplification of Cortical Surfaces of the Human Brain.
 Creator

Singleton, Lee William, Hurdal, Monica K., Kumar, Piyush, Mio, Washington, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The generation, correction, and simplification of brain surfaces from magnetic resonance imaging (MRI) data are important for studying brain characteristics, diseases, and functionality. Changes in cortical surfaces are used to compare healthy and diseased populations and they are used to understand how the brain changes as we age. We present several algorithms that use corrected MRI data to create a manifold surface, correct its topology, and simplify the resulting surface. We make...
Show moreThe generation, correction, and simplification of brain surfaces from magnetic resonance imaging (MRI) data are important for studying brain characteristics, diseases, and functionality. Changes in cortical surfaces are used to compare healthy and diseased populations and they are used to understand how the brain changes as we age. We present several algorithms that use corrected MRI data to create a manifold surface, correct its topology, and simplify the resulting surface. We make comparisons of several algorithmic choices and highlight the options that result in surfaces with the most desirable properties. In our discussion of surface generation, we present new approaches and analyze their features. We also provide a simple way to ensure that the created surface is a manifold. We compare our approaches to an existing method by examining the geometric and topological properties of the generated surfaces, including triangle count, surface area, Euler characteristic, and vertex degree. Our chapter on topology correction provides a description of our algorithm that can be used to correct the topology of a surface from the underlying volume data under a specific digital connectivity. We also present notation for new types of digital connectivities and show how our algorithm can be generalized to correct surfaces using these new connectivity schemes on the underlying volume. Our surface simplification algorithm is able to replace surface edges with new points in space rather than being restricted to the surface. We present new formulas for the fast and efficient computation of points for interior as well as boundary edges. We also provide results of several cost functions and report on their performances in surface simplification. Other algorithmic choices are also discussed and evaluated for effectiveness. We are able to produce high quality surfaces that reduce the number of surface triangles by 8586% on average while preserving surface topology, geometry, and anatomical features. On closed surfaces, our algorithm also preserves the volume inside the surface. This work provides an improvement to the general framework of surface processing. We are able to produce high quality surfaces with very few triangles and still maintain the general properties of the surface. These results have applicability to other downstream processes by reducing the processing time of applications such as flattening, inflation, and registration. Our surface results also produce much smaller files for use in future database systems. Furthermore, these algorithms can be applied to other areas of computational anatomy and scientific visualization. They have applicability to fields of medicine, computer graphics, and computational geometry.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd1702
 Format
 Thesis
 Title
 A Comparison Study of Principal Component Analysis and Nonlinear Principal Component Analysis.
 Creator

Wu, Rui, Magnan, Jerry F., Bellenot, Steven, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

In the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various...
Show moreIn the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various artificial and natural datasets sampled from: "F(x) = sin(x) + x", the Lorenz Attractor, and sunspot data. The results from the experiments have been analyzed and compared. Generally speaking, NLPCA can explain more variance than a neural network PCA (NN PCA) in lower dimensions. However, as a result of increasing the dimension, the NLPCA approximation will eventually loss its advantage. Finally, we introduce a new combination of NN PCA and NLPCA, and analyze and compare its performance.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0704
 Format
 Thesis