1,237 research outputs found
Growth and order of automorphisms of free groups and free Burnside groups
We prove that an outer automorphism of the free group is exponentially
growing if and only if it induces an outer automorphism of infinite order of
free Burnside groups with sufficiently large odd exponent.Comment: 36 pages, 4 figure
On uniformization of Burnside's curve
Main objects of uniformization of the curve are studied: its
Burnside's parametrization, corresponding Schwarz's equation, and accessory
parameters. As a result we obtain the first examples of solvable Fuchsian
equations on torus and exhibit number-theoretic integer -series for
uniformizing functions, relevant modular forms, and analytic series for
holomorphic Abelian integrals. A conjecture of Whittaker for hyperelliptic
curves and its hypergeometric reducibility are discussed. We also consider the
conversion between Burnside's and Whittaker's uniformizations.Comment: Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic
functions has been moved to arXiv:0808.348
On Jordan's measurements
The Jordan measure, the Jordan curve theorem, as well as the other generic
references to Camille Jordan's (1838-1922) achievements highlight that the
latter can hardly be reduced to the "great algebraist" whose masterpiece, the
Trait\'e des substitutions et des equations alg\'ebriques, unfolded the
group-theoretical content of \'Evariste Galois's work. The present paper
appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte
der Mathematik (1868-1942) for providing an overview of Jordan's works. On the
one hand, we shall especially investigate the collective dimensions in which
Jordan himself inscribed his works (1860-1922). On the other hand, we shall
address the issue of the collectives in which Jordan's works have circulated
(1860-1940). Moreover, the time-period during which Jordan has been publishing
his works, i.e., 1860-1922, provides an opportunity to investigate some
collective organizations of knowledge that pre-existed the development of
object-oriented disciplines such as group theory (Jordan-H\"older theorem),
linear algebra (Jordan's canonical form), topology (Jordan's curve), integral
theory (Jordan's measure), etc. At the time when Jordan was defending his
thesis in 1860, it was common to appeal to transversal organizations of
knowledge, such as what the latter designated as the "theory of order." When
Jordan died in 1922, it was however more and more common to point to
object-oriented disciplines as identifying both a corpus of specialized
knowledge and the institutionalized practices of transmissions of a group of
professional specialists
Asset Pricing Theories, Models, and Tests
An important but still partially unanswered question in the investment field is why different assets earn substantially different returns on average. Financial economists have typically addressed this question in the context of theoretically or empirically motivated asset pricing models. Since many of the proposed âriskâ theories are plausible, a common practice in the literature is to take the models to the data and perform âhorse racesâ among competing asset pricing specifications. A âgoodâ asset pricing model should produce small pricing (expected return) errors on a set of test assets and should deliver reasonable estimates of the underlying market and economic risk premia. This chapter provides an up-to-date review of the statistical methods that are typically used to estimate, evaluate, and compare competing asset pricing models. The analysis also highlights several pitfalls in the current econometric practice and offers suggestions for improving empirical tests
Investigating self-similar groups using their finite -presentation
Self-similar groups provide a rich source of groups with interesting
properties; e.g., infinite torsion groups (Burnside groups) and groups with an
intermediate word growth. Various self-similar groups can be described by a
recursive (possibly infinite) presentation, a so-called finite
-presentation. Finite -presentations allow numerous algorithms for
finitely presented groups to be generalized to this special class of recursive
presentations. We give an overview of the algorithms for finitely -presented
groups. As applications, we demonstrate how their implementation in a computer
algebra system allows us to study explicit examples of self-similar groups
including the Fabrykowski-Gupta groups. Our experiments yield detailed insight
into the structure of these groups
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