Periodic Maximal Flats Are Not Peripheral

Distribution of Periodic Torus Orbits and Duke’s Theorem for Cubic Fields

We study periodic torus orbits on space of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits; for rank 3 lattices, we show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke’s theorem about the distribution of closed geodesics on the modular surface: suitably interpre...

The Lattice of Flats and its Underlying Flag Matroid Polytope

Let M be a matroid and F the collection of all linear orderings of bases ofM. orflags of M. We define the flag matroid polytope A(F)We determine when two vertices of A(F) are adjacent, and provide a bijection between maximal chains in the lattice of flats of M and certain maximal faces of A(F).

Finiteness of Compact Maximal Flats of Bounded Volume

Let M be a complete Riemannian locally symmetric space of nonpositive curvature and of finite volume. We show that there are only finitely many compact maximal flats in M of volume bounded by a given number. As a corollary in the case M = SLn(Z)\ SLn(R)/ SOn, we give a different proof of a theorem of Remak that for any n ∈ N, there are only finitely many totally real number fields of degree n w...

Mathematical Model for Bi-objective Maximal Hub Covering Problem with Periodic Variations of Parameters

The problem of maximal hub covering as a challenging problem in operation research. Transportation programming seeks to find an optimal location of a set of hubs to reach maximum flow in a network. Since the main structure's parameters of the problem such as origin-destination flows, costs and travel time, change periodically in the real world applications, new issues arise in handling it. In t...

HECKE ORBITS OF COMPACT MAXIMAL FLATS IN ZGLn(Z)nGLn(R)=On