17,140 research outputs found
Involution and commutator length for complex hyperbolic isometries
We study decompositions of complex hyperbolic isometries as products of
involutions. We show that PU(2,1) has involution length 4 and commutator length
1, and that for all PU(,1) has involution length at most 8.Comment: 32 pages, 22 figure
Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups
We complete the classification of maximal representations of uniform complex
hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional
groups and . We prove that if is a maximal
representation of a uniform complex hyperbolic lattice , , in an exceptional Hermitian group , then and , and we describe completely the representation . The case of
classical Hermitian target groups was treated by Vincent Koziarz and the second
named author (arxiv:1506.07274). However we do not focus immediately on the
exceptional cases and instead we provide a more unified perspective, as
independent as possible of the classification of the simple Hermitian Lie
groups. This relies on the study of the cominuscule representation of the
complexification of the target group. As a by-product of our methods, when the
target Hermitian group has tube type, we obtain an inequality on the Toledo
invariant of the representation which is stronger
than the Milnor-Wood inequality (thereby excluding maximal representations in
such groups).Comment: Comments are welcome
Variational Inference for Stochastic Block Models from Sampled Data
This paper deals with non-observed dyads during the sampling of a network and
consecutive issues in the inference of the Stochastic Block Model (SBM). We
review sampling designs and recover Missing At Random (MAR) and Not Missing At
Random (NMAR) conditions for the SBM. We introduce variants of the variational
EM algorithm for inferring the SBM under various sampling designs (MAR and
NMAR) all available as an R package. Model selection criteria based on
Integrated Classification Likelihood are derived for selecting both the number
of blocks and the sampling design. We investigate the accuracy and the range of
applicability of these algorithms with simulations. We explore two real-world
networks from ethnology (seed circulation network) and biology (protein-protein
interaction network), where the interpretations considerably depends on the
sampling designs considered
Mathematical analysis of parallel convective exchangers with general lateral boundary conditions using generalized graetz modes
We propose a mathematical analysis of parallel convective exchangers for any general but longitudinally invariant domains. We analyze general Dirichlet or Neumann prescribed boundary conditions at the outer solid domain. Our study provides general mathematical expressions for the solution of convection/diffusion problems. Explicit form of generalized solutions along longitudinal coordinate are found from convoluting elementary base Graetz mode with the applied sources at the boundary. In the case of adiabatic zero flux counter-current configuration, we recover the longitudinally linearly varying solution associated with the zeroth eigenmode which can be considered as the fully developed behavior for heat-exchangers. We also provide general expression for the infinite asymptotic behavior of the solutions which depends on simple parameters such as total convective flux, outer domain perimeter and the applied boundary conditions. Practical considerations associated with the numerical precision of truncated mode decomposition is also analyzed in various configurations for illustrating the versatility of the formalism. Numerical quantities of interest are investigated, such as fluid/solid internal and external fluxes
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