Braids and riemann surfaces

Magnus, Wilhelm

doi:10.1002/cpa.3160250205

Cited by 14 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this continued investigation, we show that if a Riemann surface has genus 1, then an infinite class of non-solvable (and even simple) groups can appear and very few cyclic and non-cyclic Abelian groups can appear. When the genus is >1, our results are related to the above stated results in the following way: A necessary and sufficient condition for a covering to be Galois is that the order of the monodromy group be equal to n (the number of sheets) [16]. Thus, in this instance, the monodromy group must necessarily be represented using the right-regular representation.…”

supporting

confidence: 53%

On the Monodromy Groups of Riemann Surfaces of Genus ≧1

Kuiken

1981

Can. j. math.

View full text Add to dashboard Cite

It is well-known [5, 19] that every finite group can appear as a group of automorphisms of an algebraic Riemann surface. Hurwitz [9, 10] showed that the order of such a group can never exceed 84 (g – 1) provided that the genus g is ≧2. In fact, he showed that this bound is the best possible since groups of automorphisms of order 84 (g – 1) are obtainable for some surfaces of genus g. The problems considered by Hurwitz and others can be considered as particular cases of a more general question: Given a finite group G, what is the minimum genus of the surface for which it is a group of automorphisms? This question has been completely answered for cyclic groups by Harvey [7]. Wiman's bound 2(2g + 1), the best possible, materializes as a consequence. A further step was taken by Maclachlan who answered this question for non-cyclic Abelian groups.

show abstract

supporting

confidence: 53%

On the Monodromy Groups of Riemann Surfaces of Genus ≧1

Kuiken

1981

Can. j. math.

View full text Add to dashboard Cite

show abstract

“…The braid group B r of the plane E 2 is the subgroup of the automorphism group aut(F r ) of F r consisting of those it is enough to describe this last quotient. With the notation in Magnus [11]:…”

Section: The Modular Group Mod(γ )mentioning

confidence: 99%

The Hyperelliptic Mapping Class Group of Klein Surfaces

Bujalance

Costa

Gamboa

2001

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

In this paper we study the algebraic structure of the hyperelliptic mapping class group of Klein surfaces, which is closely related to the mapping class group of punctured discs. This group plays an important role in the study of the moduli space of hyperelliptic real algebraic curves. Our main result provides a presentation by generators and relations for the hyperelliptic mapping class group of surfaces of prescribed topological type.

show abstract

“…Assuming that the time dependent piece of the frequency is small in comparison with the constant (average) part, an expansion in powers of the former seems natural. The alternative we follow here, corresponds to writing the (formal) solution to the classical equation of motion in phase space, and implementing the Magnus expansion [7,8,9] to solve the latter. This preserves, order by order, the time evolution as a canonical transformation, both at the classical and quantum levels (since the equations of motion are linear).…”

mentioning

confidence: 99%

A Magnus approximation approach to harmonic systems with time-dependent frequencies

2018

View full text Add to dashboard Cite

We use a Magnus approximation at the level of the equations of motion for a harmonic system with a time-dependent frequency, to find an expansion for its in-out effective action, and a unitary expansion for the Bogoliubov transformation between in and out states. The dissipative effects derived therefrom are compared with the ones obtained from perturbation theory in powers of the time-dependent piece in the frequency, and with those derived using multiple scale analysis in systems with parametric resonance. We also apply the Magnus expansion to the in-in effective action, to construct reality and causal equations of motion for the external system. We show that the nonlocal equations of motion can be written in terms of a "retarded Fourier transform" evaluated at the resonant frequency. observable, namely, the dynamical equations for the external degrees of freedom. They may be obtained from the in-in effective action, and exhibit a back-reaction due to emission of quantum field modes [5,6].Except for rather special cases, it is not possible to obtain closed expressions for the effective action, even in the case of a single harmonic mode with a time-dependent frequency. Indeed, the problem of evaluating the effective action for a system like this, may be posed in terms of a functional determinant, an object which may be obtained from the solution of a linear secondorder differential equation: the classical equation of motion. Since, except for rather special cases, closed solutions for the latter are not known, it is natural to implement approximate treatments. Assuming that the time dependent piece of the frequency is small in comparison with the constant (average) part, an expansion in powers of the former seems natural. The alternative we follow here, corresponds to writing the (formal) solution to the classical equation of motion in phase space, and implementing the Magnus expansion [7,8,9] to solve the latter. This preserves, order by order, the time evolution as a canonical transformation, both at the classical and quantum levels (since the equations of motion are linear). This is a higher desirable feature when considering an expansion for the Bogoliubov transformations, Thus, our approach may be interpreted as an alternative expansion in the time-dependent part of the frequency, which preserves unitarity, and would correspond to a resummation of infinite terms on the usual expansion.The Magnus approach has been used in a large number of works and is of interest in many branches of quantum mechanics. For example, it han been used in the open quantum systems theory as a tool to investigate how to manipulate the irreversible component of open-system evolutions (decoherence and dissipation) through the application of external controllable interactions [10]. Also in the study of periodically driven systems, by means of an expansion both in the driving term and the inverse of the driving frequency, applicable to isolated or dissipative systems. In [11], a systematic Magnus expansion is used to derive explicit e...

show abstract

Braids and riemann surfaces

Cited by 14 publications

References 17 publications

On the Monodromy Groups of Riemann Surfaces of Genus ≧1

On the Monodromy Groups of Riemann Surfaces of Genus ≧1

The Hyperelliptic Mapping Class Group of Klein Surfaces

A Magnus approximation approach to harmonic systems with time-dependent frequencies

Contact Info

Product

Resources

About