No description available.

No description available.

The central result presented in this thesis is van der Waerden's theorem on arithmetic progressions. Van der Waerden's theorem guarantees that for any integers k and r, there is an n so that however the set {1, 2, ... , n} is split into r disjoint partition classes, at least one partition class will contain a k-term arithmetic progression. Presented here are a number of variations and generalizations of van der Waerden's theorem that utilize a wide range of techniques from areas of mathematics including combinatorics, number theory, algebra, and topology. / May 2007

The central result presented in this thesis is van der Waerden's theorem on arithmetic progressions. Van der Waerden's theorem guarantees that for any integers k and r, there is an n so that however the set {1, 2, ... , n} is split into r disjoint partition classes, at least one partition class will contain a k-term arithmetic progression. Presented here are a number of variations and generalizations of van der Waerden's theorem that utilize a wide range of techniques from areas of mathematics including combinatorics, number theory, algebra, and topology.

The central result presented in this thesis is van der Waerden's theorem on arithmetic progressions. Van der Waerden's theorem guarantees that for any integers k and r, there is an n so that however the set {1, 2, ... , n} is split into r disjoint partition classes, at least one partition class will contain a k-term arithmetic progression. Presented here are a number of variations and generalizations of van der Waerden's theorem that utilize a wide range of techniques from areas of mathematics including combinatorics, number theory, algebra, and topology.

Master of Arts / Department of Geography / John A. Harrington Jr / A question exists on how cultural backgrounds influence the paths students take to understand cultural geography and construct mental maps of their communities. This thesis draws on the interconnections among student multiculturalism, geo-progressions (learning trajectories in geography), and perception of the environment at the community scale. As a result of the Road Map for 21st Century Geography Education, geo-progressions have received increased attention by geography education researchers. The majority of the effort to-date has focused on the first theme of the National Geography Standards: the world in spatial terms (Standards 1-3). This study attempts to deconstruct and rethink a geo-progression by considering multiple paths to learning Geography Standard Six, "how culture and experience influence people's perceptions of places and regions." The study incorporates the concept of community, a major theme for third grade as indicated in the Kansas Standards for History, Government, and Social Studies. During this longitudinal study, students were asked to make mental maps and talk about their community-based sense of place twice during part of the 2015-2016 school year. Third-grade classrooms from four demographically distinct areas of Kansas were surveyed: Manhattan, Garden City, Horton, and Junction City. The first session was conducted in September 2015. In January 2016, the same students were asked to perform the same tasks to assess any temporal differences. Mental maps and interviews were coded and analyzed to assess the spectrum of how students perceive a spatial sense of community over time. Interviews with teachers helped document classroom-to-classroom differences in how the concept of community was incorporated into the teaching effort.

Recent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefulness of first studying the problem in a finite field environment. Using the techniques of Bourgain to give a result in other settings such as general abelian groups, the author gives a walk through, including proof, of Roth's theorem in both the one dimensional and two dimensional cases (it would be more accurate to refer to the two dimensional case as Shkredov's Theorem). In the one dimensional case the argument is at its base Meshulam's but the structure will be essentially Green's. Let Ϝⁿ [subscript p], p ≠ 2 be the finite field of cardinality N = pⁿ. For large N, any subset A ⊂ Ϝⁿ [subscript p] of cardinality ∣A ∣≳ N ∕ log N must contain a triple of the form {x, x + d, x + 2d} for x, d ∈ Ϝⁿ [subscript p], d ≠ 0. In the two dimensional case the argument is Lacey and McClain who made considerable refinements to this argument of Green who was bringing the argument to the finite field case from a paper of Shkredov. Let Ϝ ⁿ ₂ be the finite field of cardinality N = 2ⁿ. For all large N, any subset A⊂ Ϝⁿ ₂ × Ϝⁿ ₂ of cardinality ∣A ∣≳ N ² (log n) − [superscript epsilon], ε <, 1, must contain a corner {(x, y), (x + d, y), (x, y + d)} for x, y, d ∈ Ϝⁿ₂ and d ≠ 0.

L’analyse examina l’impact des relations tonales, des fonctions et des types formels transformés sur l’unité compositionnelle dans les œuvres orchestrales de Max Reger. Le contexte théorique est celui de nouvelles Formenlehre et Harmonielehre. La forme conçue comme une succession des fonctions fut analysée sur la base de la théorie des fonctions formelles de Caplin. Son apparatus analytique a été combiné avec le modèle sophistiqué de ponctuation et le concept de la déformation de la théorie de la sonate de Hepokoski et Darcy. En examinant les relations et la structure tonales, l’analyse adhère au concept de la monotonalité de Schoenberg. L’attention analytique fut focalisée sur les motifs harmoniques dérivés des accords, des progressions et de la ligne de basse. La Grundgestalt (une configuration fondamentale) fut perçue comme une structure motivique ou un contour intervallique quasi-arythmique. L’analyse montra comment Reger avait confirmé la clarté de l’unité tonale du mouvement ou de l’œuvre. / The analysis examined the impact of tonal relations and transformed formal types and functions on the compositional unity in Max Reger’s orchestral works. The theoretical background consisted of New Formenlehre and Harmonielehre. The form conceived as a succession of functions, was analyzed on the basis of Caplin’s formal function theory. His analytical apparatus was combined with the sophisticated punctuation model and the concept of “deformation” developed in the competing sonata theory of Hepokoski and Darcy. In consideration of tonal relationships and structure, the analysis adhered to Schoenberg’s concept of monotonality. The analytical attention was focused on harmonic motives derived from chords, progressions and the bass line. The Grundgestalt (basic configuration) was perceived as a motivic structure or quasi-arrhythmic interval contour. The analysis showed how Reger has confirmed the clarity of the tonal unity of a movement or work in whole.

No description available.

No description available.