Doctor of Philosophy / Mathematics / Ilia Zharkov / A smooth affine hypersurface of complex dimension n is homotopy equivalent to a real n-dimensional cell complex. We describe a recipe of constructing such cell complex for the hypersurfaces of dimension 1 and 2, i.e. for curves and surfaces. We call such cell complex a skeleton of the hypersurface. In tropical geometry, to each hypersurface, there is an associated hypersurface, called tropical hypersurface given by degenerating a family of complex amoebas. The tropical hypersurface has a structure of a polyhedral complex and it is a base of a torus fibration of the hypersurface constructed by Mikhalkin. We introduce on the edges of a tropical hypersurface an orientation given by the gradient flow of some piece-wise linear function. With the help of this orientation, we choose some sections and fibers of the fibration.These sections and fibers constitute a cell complex and we prove that this complex is the skeleton by using decomposition of the coemoeba of a classical pair-of-pants. We state and prove our main results for the case of curves and surfaces in Chapters 4 and 5.

Master of Science / Department of Mathematics / Andrew G. Bennett / An online mathematics placement exam was administered to new freshmen enrolled at Kansas State University for the Fall of 2009. The purpose of this exam is to help determine which students are prepared for a college Calculus I or Calculus II course. Problems on the exam were analyzed and grouped together using different techniques including expert analysis and item response theory to determine which problems were similar or even relevant to placement. Student scores on the exam were compared to their performance on the final exam at the end of the course as well as ACT data. This showed how well the placement exam indicated which students were prepared. A model was created using ACT information and the new information from the placement exam that improved prediction of success in a college calculus course. The new model offers a significant improvement upon what the ACT data provides to advisers. Suggestions for improvements to the test and methodology are made based upon the analysis.

Master of Science / Department of Mathematics / Marianne Korten / Distribution theory is an important tool in studying partial differential equations. Distributions are linear functionals that act on a space of smooth test functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. There are different possible choices for the space of test functions, leading to different spaces of distributions. In this report, we take a look at some basic theory of distributions and their Fourier transforms. And we also solve some typical exercises at the end.

Master of Science / Department of Mathematics / Ivan Blank / Charles N. Moore / This paper will examine some mathematical properties and models of inhomogeneous materials. By deriving models for elastic energy and heat flow we are able to establish equations that arise in the study of divergence form uniformly elliptic partial differential equations. In the late 1950's DeGiorgi and Nash showed that weak solutions to our partial differential equation lie in the Holder class. After fixing the dimension of the space, the Holder exponent guaranteed by this work depends only on the ratio of the eigenvalues. In this paper we will look at a specific geometry and show that the Holder exponent of the actual solutions is bounded away from zero independent of the eigenvalues.

Doctor of Philosophy / Department of Mathematics / David Yetter / In this paper we give a new generalization of the Khovanov homology. The construction begins with a Frobenius-algebra-like object in a category of graded vector-spaces with an anyonic braiding, with most of the relations weaken to hold only up to phase. The construction of Khovanov can be adapted to give a new link homology theory from such data. Both Khovanov's original theory and the odd Khovanov homology of Oszvath, Rassmusen and Szabo arise from special cases of the construction in which the braiding is a symmetry.

Master of Science / Department of Mathematics / Natasha Rozhkovskaya / “Incomplete open cubes” is one of the major projects of the artist Sol Lewitt. It consists of a collection of frame structures and a presentation of their diagrams. Each structure in the project is a cube with some edges removed so that the structure remains three-dimensional and connected Structures are considered to be identical if one can be transformed into another by a space rotation (but not reflection). The list of incomplete cubes consists of 122 structures. In this project, the concept of incomplete cubes was formulated in the language of graph theory. This allowed us to compare the problem posed by the artist with the similar questions of graph theory considered during the last decades. Classification of Incomplete cubes was then refined using the language of combinatorics. The list produced by the artist was then checked to be complete. And lastly, properties of Incomplete cubes in the list were studied.

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.