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Efficient Network Structures with Separable Heterogeneous Connection Costs
We introduce a heterogeneous connection model for network formation to
capture the effect of cost heterogeneity on the structure of efficient
networks. In the proposed model, connection costs are assumed to be separable,
which means the total connection cost for each agent is uniquely proportional
to its degree. For these sets of networks, we provide the analytical solution
for the efficient network and discuss stability impli- cations. We show that
the efficient network exhibits a core-periphery structure, and for a given
density, we find a lower bound for clustering coefficient of the efficient
network.Comment: 9 page
The Gauss-Manin connection on the Hodge structures
Pour tout sch\'ema simplicial complexe il existe une
application canonique , appel\'ee la
connexion de Gau\ss-Manin. Nous montrons qu'il existe une unique connexion
fonctorielle sur toute structure de Hodge-Tate mixte ayant certaines
propri\'et\'es de la connexion de Gau\ss-Manin. Cette connexion n'est pas
int\'egrable en g\'en\'eral, et alors son int\'egrabilit\'e est une condition
non triviale pour qu'une structure de Hodge soit g\'eom\'etrique.
Dans des cas particuliers, je donne des formules explicites pour la connexion
de Gau\ss-Manin sur la cohomologie singuli\`ere des vari\'et\'es alg\'ebriques
sur dans les termes de la structure de Hodge
Projective vs metric structures
We present a number of conditions which are necessary for an n-dimensional
projective structure (M,[nabla]) to include the Levi-Civita connection nabla of
some metric on M. We provide an algorithm, which effectively checks if a
Levi-Civita connection is in the projective class and, in the positive, which
finds this connection and the metric. The article also provides a basic
information on invariants of projective structures, including the treatment via
Cartan's normal projective connection. In particular we show that there is a
number of Fefferman-like conformal structures, defined on a subbundle of the
Cartan bundle of the projective structure, which encode the projectively
invariant information about (M,[nabla])
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