Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set \cFl (\cF, E) of generalized flags E$--commensurable with a fixed generalized flag$\cF$in$V$has a natural structure of an ind--variety. In the case when$V$is the standard representation of$G = SL(\infty)$, all homogeneous ind--spaces$G/P$for parabolic subgroups$P$containing a fixed splitting Cartan subgroup of$G$, are of the form$\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for$SO(\infty)$and$Sp(\infty)$. As an application of the construction, we compute the Picard group of$\cFl (\cF, E)$(and of its isotropic analogs) and show that$\cFl (\cF, E)$is a projective ind--variety if and only if$\cF$is a usual, possibly infinite, flag of subspaces in$V$

Network Model Selection Using Task-Focused Minimum Description Length

Networks are fundamental models for data used in practically every application domain. In most instances, several implicit or explicit choices about the network definition impact the translation of underlying data to a network representation, and the subsequent question(s) about the underlying system being represented. Users of downstream network data may not even be aware of these choices or their impacts. We propose a task-focused network model selection methodology which addresses several key challenges. Our approach constructs network models from underlying data and uses minimum description length (MDL) criteria for selection. Our methodology measures efficiency, a general and comparable measure of the network's performance of a local (i.e. node-level) predictive task of interest. Selection on efficiency favors parsimonious (e.g. sparse) models to avoid overfitting and can be applied across arbitrary tasks and representations. We show stability, sensitivity, and significance testing in our methodology

Network Model Selection for Task-Focused Attributed Network Inference

Networks are models representing relationships between entities. Often these relationships are explicitly given, or we must learn a representation which generalizes and predicts observed behavior in underlying individual data (e.g. attributes or labels). Whether given or inferred, choosing the best representation affects subsequent tasks and questions on the network. This work focuses on model selection to evaluate network representations from data, focusing on fundamental predictive tasks on networks. We present a modular methodology using general, interpretable network models, task neighborhood functions found across domains, and several criteria for robust model selection. We demonstrate our methodology on three online user activity datasets and show that network model selection for the appropriate network task vs. an alternate task increases performance by an order of magnitude in our experiments

Establishment of pluripotent cell lines from vertebrate species - Present status and future prospects

Pluripotent embryonic stem (ES) cells are undifferentiated cell lines derived from early embryos and are capable of unlimited undifferentiated proliferation in vitro. They retain the ability to differentiate into all cell types including germ cells in chimeric animals in vivo, and can be induced to form derivatives of all three germ layers in vitro. Mouse ES cells represent one of the most important tools in genetic research. Major applications include the targeted mutation of specific genes by homologous recombination and the discovery of new genes by gene trap strategies. These applications would be of high interest for other model organisms and also for livestock species, However, in spite of tremendous research activities, no proven ES cells colonizing the germ line have been established for vertebrate species other than mouse a nd chicken thus far. This review summarizes the current status of deriving pluripotent embryonic stem cell lines from vertebrates and recent developments in nuclear transfer technology, which may provide an alternative tool for genetic modification of livestock animals. Copyright (C) 1999 S. Karger AG, Basel

Systolic volume of homology classes

Given an integer homology class of a finitely presentable group, the systolic volume quantifies how tight could be a geometric realization of this class. In this paper, we study various aspects of this numerical invariant showing that it is a complex and powerful tool to investigate topological properties of homology classes of finitely presentable groups