2,669 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

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    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

    A Generalised abc Conjecture and Quantitative Diophantine Approximation

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    The abc Conjecture and its number field variant have huge implications across a wide range of mathematics. While the conjecture is still unproven, there are a number of partial results, both for the integer and the number field setting. Notably, Stewart and Yu have exponential abc bounds for integers, using tools from linear forms in logarithms, while Győry has exponential abc bounds in the number field case, using methods from S-unit equations [20]. In this thesis, we aim to combine these methods to give improved results in the number field case. These results are then applied to the effective Skolem-Mahler-Lech problem, and to the smooth abc conjecture. The smooth abc conjecture concerns counting the number of solutions to a+b = c with restrictions on the values of a, b and c. this leads us to more general methods of counting solutions to Diophantine problems. Many of these results are asymptotic in nature due to use of tools such as Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory". We make these lemmas effective rather than asymptotic other than on a set of size δ > 0, where δ is arbitrary. From there, we apply these tools to give an effective Schmidt’s Theorem, a quantitative Koukoulopoulos-Maynard Theorem (also referred to as the Duffin- Schaeffer Theorem), and to give effective results on inhomogeneous Diophantine Approximation on M0-sets, normal numbers and give an effective Strong Law of Large Numbers. We conclude this thesis by giving general versions of Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory"

    The Diophantine problem in Chevalley groups

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    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

    Generic multiplicative endomorphism of a field

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    We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative map, which we call ACFH. Among others, we prove that this theory is NSOP1_1 and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.Comment: 34 page

    Extension of Fujimoto's uniqueness theorems

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    Hirotaka Fujimoto considered two meromorphic maps f f and g g of Cm\mathbb{C}^m into Pn\mathbb{P}^n such that f∗(Hj)=g∗(Hj) f^*(H_j)=g^*(H_j) (1≤j≤q 1\leq j\leq q ) for q q hyperplanes Hj H_j in Pn\mathbb{P}^n in general position and proved f=g f=g under suitable conditions. This paper considers the case where f f is into Pn\mathbb{P}^n and g g is into PN\mathbb{P}^N and gives extensions of some of Fujimoto's uniqueness theorems. The dimensions N N and n n are proved to be equal under suitable conditions. New and interesting phenomena also occur

    The Generalized Montgomery Coordinate: A New Computational Tool for Isogeny-based Cryptography

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    Recently, some studies have constructed one-coordinate arithmetics on elliptic curves. For example, formulas of the xx-coordinate of Montgomery curves, xx-coordinate of Montgomery−^- curves, ww-coordinate of Edwards curves, ww-coordinate of Huff\u27s curves, ω\omega-coordinates of twisted Jacobi intersections have been proposed. These formulas are useful for isogeny-based cryptography because of their compactness and efficiency. In this paper, we define a novel function on elliptic curves called the generalized Montgomery coordinate that has the five coordinates described above as special cases. For a generalized Montgomery coordinate, we construct an explicit formula of scalar multiplication that includes the division polynomial, and both a formula of an image point under an isogeny and that of a coefficient of the codomain curve. Finally, we present two applications of the theory of a generalized Montgomery coordinate. The first one is the construction of a new efficient formula to compute isogenies on Montgomery curves. This formula is more efficient than the previous one for high degree isogenies as the 2\sqrt{\vphantom{2}}\\u27{e}lu\u27s formula in our implementation. The second one is the construction of a new generalized Montgomery coordinate for Montgomery−^- curves used for CSURF

    Elementary planes in the Apollonian orbifold

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    In this paper, we study the topological behavior of elementary planes in theApollonian orbifold MAM_A, whose limit set is the classical Apollonian gasket.The existence of these elementary planes leads to the following failure ofequidistribution: there exists a sequence of closed geodesic planes in MAM_Alimiting only on a finite union of closed geodesic planes. This contrasts withother acylindrical hyperbolic 3-manifolds analyzed in [MMO1, arXiv:1802.03853,arXiv:1802.04423]. On the other hand, we show that certain rigidity still holds: the area of anelementary plane in MAM_A is uniformly bounded above, and the union of allelementary planes is closed. This is achieved by obtaining a complete list ofelementary planes in MAM_A, indexed by their intersection with the convex coreboundary. The key idea is to recover information on a closed geodesic plane inMAM_A from its boundary data; requiring the plane to be elementary in turn putsrestrictions on these data.<br

    An abelian ambient category for behaviors in algebraic systems theory

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    We describe an abelian category ab(M)\mathbf{ab}(M) in which the solution sets of finitely many linear equations over an arbitrary ring RR with values in an arbitrary left RR-module MM reside as objects. Such solution sets are also called behaviors in algebraic systems theory. We both characterize ab(M)\mathbf{ab}(M) by a universal property and give a construction of ab(M)\mathbf{ab}(M) as a Serre quotient of the free abelian category generated by RR. We discuss features of ab(M)\mathbf{ab}(M) relevant in the context of algebraic systems theory: if RR is left coherent and MM is an fp-injective fp-cogenerator, then ab(M)\mathbf{ab}(M) is antiequivalent to the category of finitely presented left RR-modules. This provides an alternative point of view to the important module-behavior duality in algebraic systems theory. We also obtain a dual statement: if RR is right coherent and MM is fp-faithfully flat, then ab(M)\mathbf{ab}(M) is equivalent to the category of finitely presented right RR-modules. As an example application, we discuss delay-differential systems with constant coefficients and a polynomial signal space. Moreover, we propose definitions of controllability and observability in our setup.Comment: Fix typos. Add example 7.

    On subdirect products of type FPnFP_n of limit groups over Droms RAAGs

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    We generalize some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if SS is a full subdirect product of type FPs(Q)FP_s(\mathbb{Q}) of limit groups over Droms RAAGs with trivial center, then the projection of SS to the direct product of any ss of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type FPmFP_m in dimensions up to mm. In particular, this gives the values of the analytic L2L^2-Betti numbers of these groups in the respective dimensions.Comment: Accepted in Math. Proc. Cambridge Philos. So

    Ranks and approximations for families of cubic theories

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    In this paper, we study the rank characteristics for families of cubic theories, as well as new properties of cubic theories as pseudofiniteness and smooth approximability. It is proved that in the family of cubic theories, any theory is a theory of finite structure or is approximated by theories of finite structures. The property of pseudofiniteness or smoothly approximability allows one to investigate finite objects instead of complex infinite ones, or vice versa, to produce more complex ones from simple structures
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