Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solu...

We give very flexible, concrete constructions of discrete and faithful representations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and show that the criterion is satisfied in particular cases. There are direct ...

Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry rank-1 symmetric spaces prove, within each quasi-isometry class, that residually finite are abstractly commensurable. Our result gives first examples hyperbolic quasi-isometric but do not virtually geometry...

A vertex v of a graph G is domination null in G if g(v) = 0 for every minimum fractional dominating function g on G. Packing nullity is defined analogously, with reference to fractional closed neighborhood packings. We give classes of examples and examine graph operations that produce or preserve such vertices; several open problems are posed.

We show that the projective geometry PG(r − 1, q) for r > 3 is the only rank-r (combinatorial) geometry with (qr − 1)/(q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine...

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum numbe...

Learning to rank algorithms are dealing with a very large amount of features to automatically learn ranking functions, which leads to an increase of both the computational cost and the number of noisy redundant features. Feature selection is seen as a promising way to address these issues. In this paper, we propose new feature selection algorithms for learning to rank based on reweighted `2 SVM...

a trademark of IBM Inc. All other trademarks and registered trademarks are the property of their respective owners. All specifications are subject to change without notice. NOTICE OF DISCLAIMER: Xilinx is providing this design, code, or information "as is." By providing the design, code, or information as one possible implementation of this feature, application, or standard, Xilinx makes no rep...