For n = 1, the space of R-places of the rational function field R(x1, . . . , xn) is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of R-places of the rational function field R(x, y) is not metrizable. We explain how the proof generalizes to show that the space of R-places of any finitely generated formally real field extensi...

This article is concerned with generalizations of concepts like stratifiability and metrizability to arbitrary infinite cardinalities, in a way that uses linear orders in key places. This has resulted in theories which are remarkably faithful generalizations of the theories of stratifiable, metrizable, etc. spaces. For metrizable spaces, the generalization is to the class of (Tychonoff ) spaces...

In this paper, we formulate the problem of model order reduction for LTI (MIMO) dynamical systems in terms of the alignment distance. The significance of this formulation is that it establishes a natural quantitative link between the “Kalman canonical decomposition” and “model order reduction,” and fills an existing gap in this regard. The alignment distance includes a large class of state-spac...

Without the restriction of metrizability, topological dynamical systems (X, 〈Ts〉s∈G) are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characteri...

Two compact Hausdorff spaces are co-elementarily equivalent if they have homeomorphic ultracopowers; equivalently if their Banach spaces of continuous real-valued functions have isometrically isomorphic Banach ultrapowers (or, approximately satisfy the same positive-bounded sentences). We prove here that any locally connected compact metrizable space co-elementarily equivalent with an arc (resp...

ing the above, for a (not necessarily countable) family . . . φ2 // B1 φ1 // Bo of Banach spaces with continuous linear transition maps as indicated, not recessarily requiring the continuous linear maps to be injective (or surjective), a (projective) limit limiBi is a topological vector space with continuous linear maps limiBi → Bj such that, for every compatible family of continuous linear map...