13,543 research outputs found
The torsion group of endotrivial modules
Let G be a finite group and let T(G) be the abelian group of equivalence
classes of endotrivial kG-modules, where k is an algebraically closed field of
characteristic p. We determine, in terms of the structure of G, the kernel of
the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, in
the case when S is abelian. This provides a classification of all torsion
endotrivial kG-modules in that case
The classification of torsion endo-trivial modules
This paper is a major step in the classification of endotrivial modules over
p-groups. Let G be a finite p-group and k be a field of characteristic p. A
kG-module M is an endo-trivial module if {\End_k(M)\cong k\oplus F} as
kG-modules, where F is a free module. The classification of endo-trivial
modules is the crucial step for understanding the more general class of
endo-permutation modules. The endo-permutation modules play an important role
in module theory, in particular as source modules, and in block theory where
they appear in the description of source algebras. Endo-trivial modules are
also important in the study of both derived equivalences and stable
equivalences of group algebras and block algebras.
The collection of isomorphism classes of endo-trivial modules modulo
projectives is an abelian group under tensor product. The main result of this
paper is that this group is torsion free except in the case that G is cyclic,
quaternion or semi-dihedral. Hence for any p-group which is not cyclic,
quaternion or semi-dihedral and any finitely generated kG-module M, if M
\otimes_k M \otimes_k ... \otimes_k M \cong k \oplus P for some projective
module P and some finite number of tensor products, then M \cong k \oplus Q for
some projective module Q. The proof uses a reduction to the cases in which G is
an extraspecial or almost extraspecial p-group, proved in a previous paper of
the authors, and makes extensive use of the theory of support varieties for
modules.Comment: 61 pages, published versio
Finite element simulation of the liquid-liquid transition to metallic hydrogen
Hydrogen at high temperature and pressure undergoes a phase transition from a
liquid molecular phase to a conductive atomic state, or liquid metallic
hydrogen, sometimes referred to as the plasma phase transition (PPT). The PPT
phase line was observed in a recent experiment studying laser-pulse heated
hydrogen in a diamond anvil cell in the pressure range for temperatures up to . The experimental
signatures of the transition are (i) a negative pressure-temperature slope,
(ii) a plateau in the heating curve, assumed to be related to the latent heat
of transformation, and (iii) an abrupt increase in the reflectance of the
sample. We present a finite element simulation that accurately takes into
account the position and time dependence of the heat deposited by the laser
pulse. We calculate the heating curves and the sample reflectance and
transmittance. This simulation confirms that the observed plateaus are related
to the phase transition, however we find that large values of latent heat are
needed and may indicate that dynamics at the transition are more complex than
considered in current models. Finally, experiments are proposed that can
distinguish between a change in optical properties due to a transition to a
metallic state or due to closure of the bandgap in molecular hydrogen.Comment: 23 pages, 4 figure
Central Limit Theorem for a Class of Relativistic Diffusions
Two similar Minkowskian diffusions have been considered, on one hand by
Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR], [DR]),
and on the other hand by Dunkel and H\"anggi ([DH1], [DH2]). We address here
two questions, asked in [DR] and in ([DH1], [DH2]) respectively, about the
asymptotic behaviour of such diffusions. More generally, we establish a central
limit theorem for a class of Minkowskian diffusions, to which the two above
ones belong. As a consequence, we correct a partially wrong guess in [DH1].Comment: 20 page
Agglomeration externalities, innovation and regional growth: Theoretical perspectives and meta-analysis
Technological change and innovation and are central to the quest for regional development. In the globally-connected knowledge-driven economy, the relevance of agglomeration forces that rely on proximity continues to increase, paradoxically despite declining real costs of information, communication and transportation. Globally, the proportion of the population living in cities continues to grow and sprawling cities remain the engines of regional economic transformation. The growth of cities results from a complex chain that starts with scale, density and geography, which then combine with industrial structure characterised by its extent of specialisation, competition and diversity, to yield innovation and productivity growth that encourages employment expansion, and further urban growth through inward migration. This paper revisits the central part of this virtuous circle, namely the Marshall-Arrow-Romer externalities (specialisation), Jacobs externalities (diversity) and Porter externalities (competition) that have provided alternative explanations for innovation and urban growth. The paper evaluates the statistical robustness of evidence for such externalities presented in 31 scientific articles, all building on the seminal work of Glaeser et al. (1992). We aim to explain variation in estimation results using study characteristics by means of ordered probit analysis. Among the results, we find that the impact of diversity depends on how it is measured and that diversity is important for the high-tech sector. High population density increases the chance of finding positive effects of specialisation on growth. More recent data find more positive results for both specialization and diversity, suggesting that agglomeration externalities become more important over time. Finally, primary study results depend on whether or not the externalities are considered jointly and on other features of the regression model specification
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