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Articles 1 - 10 of 10
Full-Text Articles in Number Theory
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN where N≥ 2. Let ZK be a union of varieties defined over K such that V ⊈ ZK. We prove the existence of a point of small height in V \ ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing ZK, where dependence on …
The Four-Color Theorem And Chromatic Numbers Of Graphs, Sarah E. Cates
The Four-Color Theorem And Chromatic Numbers Of Graphs, Sarah E. Cates
Undergraduate Theses and Capstone Projects
We study graph colorings of the form made popular by the four-color theorem. Proved by Appel and Haken in 1976, the Four-Color Theorem states that all planar graphs can be vertex-colored with at most four colors. We consider an alternate way to prove the Four-Color Theorem, introduced by Hadwiger in 1943 and commonly know as Hadwiger’s Conjecture. In addition, we examine the chromatic number of graphs which are not planar. More specifically, we explore adding edges to a planar graph to create a non-planar graph which has the same chromatic number as the planar graph which we started from.
General Wp-Bailey Chains, James Mclaughlin, Peter Zimmer
General Wp-Bailey Chains, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
Motivated by a recent paper of Liu and Ma, we describe a number of general WP-Bailey chains. We show that many of the existing WP-Bailey chains (or branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and Liu and Ma, arise as special cases of these general WP-Bailey chains. We exhibit three new branches of the WP-Bailey tree, branches which also follow as special cases of these general WP-Bailey chains. Finally, we describe a number of new transformation formulae for basic hypergeometric series which arise as consequences of these new WP-Bailey chains.
Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
Mathematics Faculty Publications
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers Ramanujan-type identities.
An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin
An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin
Mathematics Faculty Publications
Ramanujan stated an identity to the effect that if three sequences {an}, {bn} and {cn} are defined by r1(x) =: ∑∞ n=0 anx n , r2(x) =: ∑∞ n=0 bnx n and r3(x) =: ∑∞ n=0 cnx n (here each ri(x) is a certain rational function in x), then a 3 n + b 3 n − c 3 n = (−1)n , ∀ n ≥ 0. Motivated by this amazing identity, we state and prove a more general identity involving eleven sequences, the new identity being ”more general” in the sense that equality holds not just for the power …
Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin
Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin
Mathematics Faculty Publications
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.
Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
If k is set equal to aq in the definition of a WP Bailey pair, βn(a, k) = Xn j=0 (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), this equation reduces to βn = Pn j=0 αj . This seemingly trivial relation connecting the αn’s with the βn’s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for poly-basic hypergeometric series.
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
CMC Senior Theses
Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.
Super Special Codes Using Super Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Super Special Codes Using Super Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Branch Mathematics and Statistics Faculty and Staff Publications
The new classes of super special codes are constructed in this book using the specially constructed super special vector spaces. These codes mainly use the super matrices. These codes can be realized as a special type of concatenated codes. This book has four chapters. In chapter one basic properties of codes and super matrices are given. A new type of super special vector space is constructed in chapter two of this book. Three new classes of super special codes namely, super special row code, super special column code and super special codes are introduced in chapter three. Applications of these …
The Fibonacci Sequence, Arik Avagyan
The Fibonacci Sequence, Arik Avagyan
A with Honors Projects
A review was made of the Fibonacci sequence, its characteristics and applications.