6,310 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

    Get PDF
    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

    Curves with few bad primes over cyclotomic Z_l extensions

    Get PDF

    The Diophantine problem in Chevalley groups

    Full text link
    In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R)G_\pi (\Phi,R), where Φ\Phi is an indecomposable root system of rank >1> 1, RR is an arbitrary commutative ring with 11. We establish a variant of double centralizer theorem for elementary unipotents xα(1)x_\alpha(1). This theorem is valid for arbitrary commutative rings with 11. The result is principle to show that any one-parametric subgroup XαX_\alpha, α∈Φ\alpha \in \Phi, is Diophantine in GG. Then we prove that the Diophantine problem in Gπ(Φ,R)G_\pi (\Phi,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in RR. 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

    Full text link
    Let GG be a countable discrete group. We give a necessary and sufficient condition for a transitive GG-system to be disjoint with all minimal GG-systems, which implies that if a transitive GG-system is disjoint with all minimal GG-systems, then it is ∞\infty-transitive, i.e. (Xk,G)(X^k,G) is transitive for all k∈Nk\in\N, and has dense minimal points. In addition, we show that any ∞\infty-transitive GG-system with dense distal points are disjoint with all minimal GG-systems.Comment: 33page

    2023-2024 Catalog

    Get PDF
    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

    Positivity Problems for Reversible Linear Recurrence Sequences

    Get PDF

    New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations

    Full text link
    The Hamiltonians of finite type discrete quantum mechanics with real shifts are real symmetric matrices of order N+1N+1. We discuss the Darboux transformations with higher degree (>N>N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after MM-steps are of order N+M+1N+M+1. Based on twelve orthogonal polynomials ((qq-)Racah, (dual, qq-)Hahn, Krawtchouk and five types of qq-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 (≤N\leq N) 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

    Get PDF
    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

    Full text link
    For an algebraic number α\alpha of degree nn, let Mα\mathcal{M}_{\alpha} be the Z\mathbb{Z}-module generated by 1,α,…,αn−11,\alpha ,\ldots ,\alpha^{n-1}; then Zα:={ξ∈Q(α): ξMα⊆Mα}\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\, \xi\mathcal{M}_{\alpha}\subseteq\mathcal{M}_{\alpha}\} is the ring of scalars of Mα\mathcal{M}_{\alpha}. We call an order of the shape Zα\mathbb{Z}_{\alpha} \emph{rationally monogenic}. If α\alpha is an algebraic integer, then Zα=Z[α]\mathbb{Z}_{\alpha}=\mathbb{Z}[\alpha ] is monogenic. Rationally monogenic orders are special types of invariant orders of binary forms, which have been studied intensively. If α,β\alpha ,\beta are two GL2(Z)\text{GL}_2(\mathbb{Z})-equivalent algebraic numbers, i.e., β=(aα+b)/(cα+d)\beta =(a\alpha +b)/(c\alpha +d) for some (abcd)∈GL2(Z)\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\in\text{GL}_2(\mathbb{Z}), then Zα=Zβ\mathbb{Z}_{\alpha}=\mathbb{Z}_{\beta}. Given an order O\mathcal{O} of a number field, we call a GL2(Z)\text{GL}_2(\mathbb{Z})-equivalence class of α\alpha with Zα=O\mathbb{Z}_{\alpha}=\mathcal{O} a \emph{rational monogenization} of O\mathcal{O}. We prove the following. If KK is a quartic number field, then KK has only finitely many orders with more than two rational monogenizations. This is best possible. Further, if KK is a number field of degree ≥5\geq 5, the Galois group of whose normal closure is 55-transitive, then KK 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 SS-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
    • …
    corecore