2,669 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
A Generalised abc Conjecture and Quantitative Diophantine Approximation
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
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
Generic multiplicative endomorphism of a field
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 NSOP 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
Hirotaka Fujimoto considered two meromorphic maps and of
into such that () for hyperplanes in in general position
and proved under suitable conditions. This paper considers the case
where is into and is into and gives
extensions of some of Fujimoto's uniqueness theorems. The dimensions and
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
Recently, some studies have constructed one-coordinate arithmetics on elliptic curves. For example, formulas of the -coordinate of Montgomery curves, -coordinate of Montgomery curves, -coordinate of Edwards curves, -coordinate of Huff\u27s curves, -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 \\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
In this paper, we study the topological behavior of elementary planes in theApollonian orbifold , 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 limiting 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 is uniformly bounded above, and the union of allelementary planes is closed. This is achieved by obtaining a complete list ofelementary planes in , indexed by their intersection with the convex coreboundary. The key idea is to recover information on a closed geodesic plane in 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
We describe an abelian category in which the solution sets
of finitely many linear equations over an arbitrary ring with values in an
arbitrary left -module reside as objects. Such solution sets are also
called behaviors in algebraic systems theory. We both characterize
by a universal property and give a construction of
as a Serre quotient of the free abelian category generated by
. We discuss features of relevant in the context of
algebraic systems theory: if is left coherent and is an fp-injective
fp-cogenerator, then is antiequivalent to the category of
finitely presented left -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 is right coherent and is fp-faithfully
flat, then is equivalent to the category of finitely presented
right -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 of limit groups over Droms RAAGs
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 is a full subdirect
product of type of limit groups over Droms RAAGs with
trivial center, then the projection of to the direct product of any 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 in dimensions up to . In particular, this gives the values of
the analytic -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
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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