556 research outputs found
Euler Characteristics of Categories and Homotopy Colimits
In a previous article, we introduced notions of finiteness obstruction, Euler
characteristic, and L^2-Euler characteristic for wide classes of categories. In
this sequel, we prove the compatibility of those notions with homotopy colimits
of I-indexed categories where I is any small category admitting a finite
I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit
Formula include formulas for products, homotopy pushouts, homotopy orbits, and
transport groupoids. We also apply our formulas to Haefliger complexes of
groups, which extend Bass--Serre graphs of groups to higher dimensions. In
particular, we obtain necessary conditions for developability of a finite
complex of groups from an action of a finite group on a finite category without
loops.Comment: 44 pages. This final version will appear in Documenta Mathematica.
Remark 8.23 has been improved, discussion of Grothendieck construction has
been slightly expanded at the beginning of Section 3, and a few other minor
improvements have been incoporate
Finiteness obstructions and Euler characteristics of categories
We introduce notions of finiteness obstruction, Euler characteristic,
L^2-Euler characteristic, and M\"obius inversion for wide classes of
categories. The finiteness obstruction of a category Gamma of type (FP) is a
class in the projective class group K_0(RGamma); the functorial Euler
characteristic and functorial L^2-Euler characteristic are respectively its
RGamma-rank and L^2-rank. We also extend the second author's K-theoretic
M\"obius inversion from finite categories to quasi-finite categories. Our main
example is the proper orbit category, for which these invariants are
established notions in the geometry and topology of classifying spaces for
proper group actions. Baez-Dolan's groupoid cardinality and Leinster's Euler
characteristic are special cases of the L^2-Euler characteristic. Some of
Leinster's results on M\"obius-Rota inversion are special cases of the
K-theoretic M\"obius inversion.Comment: Final version, accepted for publication in the Advances in
Mathematics. Notational change: what was called chi(Gamma) in version 1 is
now called chi(BGamma), and chi(Gamma) now signifies the sum of the
components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25
summarizes when all Euler characteristics are equal. Minor typos have been
corrected. 88 page
The Ore condition, affiliated operators, and the lamplighter group
Let G be the wreath product of Z and Z/2, the so called lamplighter group and
k a commutative ring. We show that kG does not have a classical ring of
quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka
notebook problem. Assume that kG is contained in a ring R in which the element
1-x is invertible, with x a generator of Z considered as subset of G. Then R is
not flat over kG. If k is the field of complex numbers, this applies in
particular to the algebra UG of unbounded operators affiliated to the group von
Neumann algebra of G. We present two proofs of these results. The second one is
due to Warren Dicks, who, having seen our argument, found a much simpler and
more elementary proof, which at the same time yielded a more general result
than we had originally proved. Nevertheless, we present both proofs here, in
the hope that the original arguments might be of use in some other context not
yet known to us.Comment: LaTex2e, 7 pages. Added a new proof of the main result (due to Warren
Dicks) which is shorter, easier and more elementary, and at the same time
yields a slightly more general result. Additionally: misprints removed. to
appear in Proceedings of "Higher dimensional manifold theory", Conference at
ICTP Trieste 200
Finite group extensions and the Baum-Connes conjecture
In this note, we exhibit a method to prove the Baum-Connes conjecture (with
coefficients) for extensions with finite quotients of certain groups which
already satisfy the Baum-Connes conjecture. Interesting examples to which this
method applies are torsion-free finite extensions of the pure braid groups,
e.g. the full braid groups, or certain fundamental groups of complements of
links in S^3.Comment: AMS-Latex, logical structure clarified, final version, to appear in
Geometry and Topolog
Berufserfolg Bamberger Soziologen: Ergebnisse der ersten Bamberger Absolventenstudie
Die Autoren berichten über die Ergebnisse einer Studie über den Berufsverlauf von Bamberger HochschulabsolventInnen, die im Zeitraum 1997/98 erstmals vom Lehrstuhl für Soziologie I durchgeführt wurde. Dargestellt wird zum einen die Berufseinmündung und die aktuelle Beschäftigungssituation der insgesamt 116 Soziologie-AbsolventInnen, die an der schriftlichen Befragung teilnahmen. Zum anderen werden die Einflussfaktoren auf die Berufseinmündung und den Berufserfolg beschrieben, die sich bei den sozio-demographischen Variablen, den studieninternen und -externen Qualifikationen, bei der Wahl der Studienschwerpunkte und beim Erreichen individuell gesetzter Studienziele erkennen lassen. Für die verschiedenen Studienschwerpunkte innerhalb der Soziologie bzw. für die verschiedenen anvisierten Berufsfelder lassen sich ferner unterschiedliche Karrierechancen ausmachen. Von den für Bamberg charakteristischen Berufsfeldern scheinen die PR-Branche und die wissenschaftliche Karriere die günstigsten Vorzeichen zu setzen. (ICI
Finiteness obstructions and Euler characteristics of categories
"Vegeu el resum a l'inici del document del fitxer adjunt"
Characterizing degradation gradients through land cover change analysis in rural Eastern Cape, South Africa
CITATION: Munch, Z., et al. 2017. Characterizing degradation gradients through land cover change analysis in rural Eastern Cape, South Africa. Geosciences, 7(1):7, doi:10.3390/geosciences7010007.The original publication is available at http://www.mdpi.comLand cover change analysis was performed for three catchments in the rural Eastern Cape, South Africa, for two time steps (2000 and 2014), to characterize landscape conversion trajectories for sustained landscape health. Land cover maps were derived: (1) from existing data (2000); and (2) through object-based image analysis (2014) of Landsat 8 imagery. Land cover change analysis was facilitated using land cover labels developed to identify landscape change trajectories. Land cover labels assigned to each intersection of the land cover maps at the two time steps provide a thematic representation of the spatial distribution of change. While land use patterns are characterized by high persistence (77%), the expansion of urban areas and agriculture has occurred predominantly at the expense of grassland. The persistence and intensification of natural or invaded wooded areas were identified as a degradation gradient within the landscape, which amounted to almost 10% of the study area. The challenge remains to determine significant signals in the landscape that are not artefacts of error in the underlying input data or scale of analysis. Systematic change analysis and accurate uncertainty reporting can potentially address these issues to produce authentic output for further modelling.http://www.mdpi.com/2076-3263/7/1/7Publisher's versio
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