6,310 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
The Diophantine problem in Chevalley groups
In this paper we study the Diophantine problem in Chevalley groups , where is an indecomposable root system of rank , is
an arbitrary commutative ring with . We establish a variant of double
centralizer theorem for elementary unipotents . This theorem is
valid for arbitrary commutative rings with . The result is principle to show
that any one-parametric subgroup , , is Diophantine
in . Then we prove that the Diophantine problem in is
polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine
problem in . This fact gives rise to a number of model-theoretic corollaries
for specific types of rings.Comment: 44 page
Disjointness with all minimal systems under group actions
Let be a countable discrete group. We give a necessary and sufficient
condition for a transitive -system to be disjoint with all minimal
-systems, which implies that if a transitive -system is disjoint with all
minimal -systems, then it is -transitive, i.e. is
transitive for all , and has dense minimal points. In addition, we show
that any -transitive -system with dense distal points are disjoint
with all minimal -systems.Comment: 33page
2023-2024 Catalog
The 2023-2024 Governors State University Undergraduate and Graduate Catalog is a comprehensive listing of current information regarding:Degree RequirementsCourse OfferingsUndergraduate and Graduate Rules and Regulation
New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
The Hamiltonians of finite type discrete quantum mechanics with real shifts
are real symmetric matrices of order . We discuss the Darboux
transformations with higher degree () polynomial solutions as seed
solutions. They are state-adding and the resulting Hamiltonians after -steps
are of order . Based on twelve orthogonal polynomials ((-)Racah,
(dual, -)Hahn, Krawtchouk and five types of -Krawtchouk), new finite type
multi-indexed orthogonal polynomials are obtained, which satisfy second order
difference equations, and all the eigenvectors of the deformed Hamiltonian are
described by them. We also present explicit forms of the Krein-Adler type
multi-indexed orthogonal polynomials and their difference equations, which are
obtained from the state-deleting Darboux transformations with lower degree
() polynomial solutions as seed solutions.Comment: 50 pages. Typos are corrected. To appear in PTE
Student activities in solving mathematics problems with a computational thinking using Scratch
The progress of the times requires students to be able to think quickly. Student activities in learning are always associated with technology and students’ thinking activities and are expected to think computationally. Therefore, this study aimed to determine how learning with the concept of computational thinking (CT) using the Scratch program can improve students’ mathematical problem-solving abilities. An exploratory research design was conducted by involving 132 grade VIII students in Kuningan, Indonesia. Data analysis began with organization, data description, and statistical testing. The results showed that students performed the concepts of abstraction thinking, algorithmic thinking, decomposition, and evaluation in solving mathematical problems. There were differences in students’ problem-solving abilities before and after the intervention. Students’ activeness in solving problems using the CT concept through a calculator significantly affected 52.3% of the ability to solve mathematical problems
Orders with few rational monogenizations
For an algebraic number of degree , let be
the -module generated by ; then
is the ring of scalars
of . We call an order of the shape
\emph{rationally monogenic}. If is an algebraic integer, then
is monogenic. Rationally monogenic
orders are special types of invariant orders of binary forms, which have been
studied intensively. If are two
-equivalent algebraic numbers, i.e., for some
,
then . Given an order of
a number field, we call a -equivalence class of
with a \emph{rational
monogenization} of .
We prove the following. If is a quartic number field, then has only
finitely many orders with more than two rational monogenizations. This is best
possible. Further, if is a number field of degree , the Galois
group of whose normal closure is -transitive, then has only finitely
many orders with more than one rational monogenization. The proof uses
finiteness results for unit equations, which in turn were derived from
Schmidt's Subspace Theorem. We generalize the above results to rationally
monogenic orders over rings of -integers of number fields. Our results
extend work of B\'{e}rczes, Gy\H{o}ry and the author from 2013 on multiply
monogenic orders.Comment: This is the final version which has been published on-line by Acta
Arithmetica. It is a slight modification of the previous versio
- …