In this note we obtain necessary and sufficient conditions for a convergence space to have a smallest Hausdorff compactification and to have a smallest regular compactification. Introduction. A Hausdorff convergence space as defined in [1] always has a Stone-Cech compactification which can be obtained by a slight modification of the result in [3]. But in general this need not be the largest Hau...

A compactification of the set of conditioned invariant subspaces of fixed dimension for an observable pair (C, A) is proposed. It contains the almost conditioned invariant subspaces of the same dimension. In certain cases the compactification is shown to be smooth and a complete geometric description is given in the case of a single output system.

Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any mdimensional ...

Let G = Sp2g be the symplectic group (Chevalley group scheme over Z). Let K be a maximal compact subgroup of G = G(R). The symmetric space G(R)/K is commonly referred to as the Siegel upper half plane, we denote it by hg. It is common to realise hg = {A ∈ Mat g(C) | tA = A,=(A) > 0}. Thus it is a complex analytic space. Let Γ = G(Z) (respectively a congruence subgroup of). Then it is well known...

When asked about the ten-dimensional nature of superstring theory, Richard Feynman once replied, “The only prediction [string theory] makes is one that has to be explained away because it doesn’t agree with experiment.” [8] In a sense, this skepticism is not unfounded, and if string theory did not have so many other desirable features (being a renormalizable theory containing interacting spin-2...

Conventional wisdom states that Newton’s force law implies only four non-compact dimensions. We demonstrate that this is not necessarily true in the presence of a non-factorizable background geometry. The specific example we study is a single 3-brane embedded in five dimensions. We show that even without a gap in the Kaluza-Klein spectrum, four-dimensional Newtonian and general relativistic gra...

We study the question when a ∗-autonomous (Mix-)category has a representation as a ∗-autonomous category of a compact one. We prove that necessary and sufficient condition is that weak distributivity maps are monic (or, equivalently epic). For a Mix-category, this condition is, in turn, equivalent to the requirement that Mix-maps be monic (or epic). We call categories satisfying this property t...

We consider the compactification of Matrix theory on tori with background anti-symmetric tensor field. Douglas and Hull have recently discussed how noncommutative geometry appears on the tori. In this paper, we demonstrate the concrete construction of this compactification of Matrix theory in a similar way to that previously given by Taylor.

This is part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in part I, a compactification of these isometry groups, and called “bipolarized” those Lorentz manifolds having a “trivial ” compactification. Here we show a geometric rigidity of non-bipolarized Lorentz manifolds; that is, they are (at least locally) warped products of constant curvature Lorentz ...

A non-zero constant Jacobian polynomial map F = (P,Q) : C −→ C 2 has a polynomial inverse if the component P is a simple polynomial, i.e. if, when P extended to a morphism p : X −→ P of a compactification X of C, the restriction of p to each irreducible component C of the compactification divisor D = X −C is either degree 0 or 1.