80,957 research outputs found
Lenstra-Hurwitz Cliques In Real Quadratic Fields
Let be a number field and let \OO_K denote its ring of integers. We can define a graph whose vertices are the elements of \OO_K such that an edge exists between two algebraic integers if their difference is in the units \OO_K^{\times}. Lenstra showed that the existence of a sufficiently large clique (complete subgraph) will imply that the ring \OO_K is Euclidean with respect to the field norm. A recent generalization of this work tells us that if we draw more edges in the graph, then a sufficiently large clique will imply the weaker (but still very interesting) conclusion that has class number one.
This thesis aims to understand this new result and produce further examples of cliques in rings of integers. Lenstra, Long, and Thistlethwaite analyzed cliques and gave us class number one through a prime element. We were able to extend and generalize their result to larger cliques through prime power elements while still preserving our desired property of class number one. Our generalization gave us that class number one is preserved if the number field contained a clique that is generated by a prime power
Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces
Two applications of Nash-Williams' theory of barriers to sequences on Banach
spaces are presented: The first one is the -saturation of ,
countable compacta. The second one is the construction of weakly-null sequences
generalizing the example of Maurey-Rosenthal
Fast and stable contour integration for high order divided differences via elliptic functions
In this paper, we will present a new method for evaluating high order divided
differences for certain classes of analytic, possibly, operator valued
functions. This is a classical problem in numerical mathematics but also
arises in new applications such as, e.g., the use of generalized convolution
quadrature to solve retarded potential integral equations. The functions which
we will consider are allowed to grow exponentially to the left complex half
plane, polynomially to the right half plane and have an oscillatory behaviour
with increasing imaginary part. The interpolation points are scattered in a
large real interval. Our approach is based on the representation of divided
differences as contour integral and we will employ a subtle parameterization
of the contour in combination with a quadrature approximation by the
trapezoidal rule
Identification of large masses of citrus fruit and rice fields in eastern Spain
There are no author-identified significant results in this report
- …