Browsing by Tema "Abelian group"

Now showing 1 - 11 of 11

A Characterization of Strongly Dependent Ordered Abelian Groups
(National University of Colombia, 2018) Alfred Dolich; John Goodrick
We characterize all ordered Abelian groups whose ﬁrst-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].
A SIXTH-ORDER CALCULATION OF THE HIGH-ENERGY BEHAVIOUR OF THE FORM-FACTORS IN NON-ABELIAN GAUGE THEORIES
(1976) Yunfeng Ai
The high-energy behaviour of the Fermion form-factors in non-Abelian theories is calculated to the sixth order in the perturbation theory. Certain manipulation methods and criterions for checking the results are found to guarantee that, when domains of integration having leading contributions are separated out, no double-counting or omission is made. Results of the calculation show that, to the leading order in logarithms, the gauge group scalier part of the form-factor exponentiates, while the gauge group vector part does not. The breaking of the exponentiation of the latter begins at the sixth order.
Amalgamation functors and boundary properties in simple theories
(Cornell University, 2010) John Goodrick; Byunghan Kim; Alexei Kolesnikov
This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various "dimensions" n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context.
Digital representation of semigroups and groups
(Springer Science+Business Media, 2008) Stefano Ferri; Neil Hindman; Dona Strauss
DISCRETE SETS DEFINABLE IN STRONG EXPANSIONS OF ORDERED ABELIAN GROUPS
(Cambridge University Press, 2024) Alfred Dolich; John Goodrick
Abstract We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a particular emphasis on the set $D'$ comprised of differences between successive elements. In particular, if the burden of the structure is at most n , then the result of applying the operation $D \mapsto D'\ n$ times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden $2$ , we show that any definable unary discrete set must be definable in some elementary extension of the structure $\langle \mathbb{R}; <, +, \mathbb{Z} \rangle $ (Theorem 1.3).
Divisibility in certain automorphism groups
(Springer Nature, 2007) Ramiro Lafuente-Rodriguez
We study solvability of equations of the form x n = g in the groups of order automorphisms of archimedean-complete totally ordered groups of rank 2. We determine exactly which automorphisms of the unique abelian such group have square roots, and we describe all automorphisms of the general ones.
Fusion of 2-elements in groups of finite Morley rank
(Cambridge University Press, 2001) Luis-Jaime Corredor
The Alperin-Goldschmidt Fusion Theorem [1, 5], when combined with pushing up [7], was a useful tool in the classification of the finite simple groups. Similar theorems are needed in the study of simple groups of finite Morley rank, in the even type case (that is, when the Sylow 2-subgroups are of bounded exponent, as in algebraic groups over fields of characteristic 2). In that context a body of results relating to fusion of 2-elements and the structure of 2-local subgroups is needed: pushing up, and the classification of groups with strongly or weakly embedded subgroups, or have strongly closed abelian subgroups (c.f, [2]). Two theorems of Alperin-Goldschmidt type are proved here. Both are needed in applications. The following is an exact analog of the Alperin-Goldschmidt Fusion Theorem for groups of finite Morley rank, in the case of 2-elements: Theorem 1.1. Let G be a group of finite Morley rank, and P a Sylow 2- subgroup of G . If A, B ⊆ P are conjugate in G, then there are subgroups H i ≤ P and elements x i ∈ N ( H i ) for 1 ≤ i ≤ n , and an element y ∈ N ( P ), such that for all i : 1. H i is a tame intersection of two Sylow 2- subgroups ; 2. C P ( H i ) ≤ H i ; 3. N ( H i )/ H i is 2- isolated and (a) (b) .
Galois coverings of moduli spaces of curves and loci of curves with symmetry
(Cornell University, 2011) Marco Boggi
Let $\ccM_{g,[n]}$, for $2g-2+n>0$, be the stack of genus $g$, stable algebraic curves, endowed with $n$ unordered marked points. Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack $\ccM_{g}$. In §2 of this paper, we introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of $\ccM_{g,[n]}$ equivalent to the tower of all geometric level structures over $\ccM_{g,[n]}$. In §3, Looijenga level structures are interpreted geometrically in terms of moduli of curves with symmetry. A byproduct of this characterization is a simple criterion for their smoothness. As a consequence of this criterion, it is shown that Looijenga level structures are smooth under mild hypotheses. The second part of the paper, from §4, deals with the problem of describing the D-M boundary of level structures. In §6, a description is given of the nerve of the D-M boundary of abelian level structures. In §7, it is shown how this construction can be used to "approximate" the nerve of Looijenga level structures. These results are then applied to elaborate a new approach to the congruence subgroup problem for the Teichmüller modular group.
Galois coverings of moduli spaces of curves and loci of curves with symmetry
(Springer Science+Business Media, 2013) Marco Boggi
Multiplicity theory of projections in abelian von neumann algebras
(1988) T. V. Panchapagesan
La teoría de multiplicidad espectral se generaliza para las proyecciones en un algebra conmutativa de von Neumann. La bien conocida descomposición de un álgebra de tipo I de von Neumann en una suma directa de álgebras de tipo In se deduce como una consecuencia
Trivializing group actions on braided crossed tensor categories and graded braided tensor categories
(Mathematical Society of Japan, 2022) César Galíndo
For an abelian group $A$, we study a close connection between braided $A$-crossed tensor categories with a trivialization of the $A$-action and $A$-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a tensor category $\mathcal{C}$ is given by an element $O(T) \in H^2(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T) = 0$, the set of obstructions forms a torsor over $\operatorname{Hom}(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$, where $\operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed Etingof et al., allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided $A$-crossed tensor category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $\mathbb{Z}/2\mathbb{Z}$-crossed structures over Tambara–Yamagami fusion categories and, consequently, a conceptual interpretation of the results by Siehler about the classification of braidings over Tambara–Yamagami categories.