James J. Madden

Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings

Preordered and partially ordered rings.- Reflective subcategories.- Totally ordered and real closed fields.- Real spectra of preordered rings.- Epimorphisms of reduced porings.- Functions and representable porings.- Semi-algebraic functions.- Comparing reflectors.- Constructing reflectors.- H-closed epireflectors.- Quotient-closed reflectors.- The real closure reflector.- Arities of reflectors and approximations by H-closed reflectors.- Epimorphic extensions of reduced porings.- Essential monoreflectors.- Reflections of totally ordered fields.- von Neumann regular f-rings.- Totally ordered domains.- Reduced f-rings.- Rings of continuous piecewise polynomial functions.- Rings of continuous piecewise rational functions.- Discontinuous semi-algebraic functions.- The lattice of H-closed monoreflectors.

A Note on Generic Properties of Continuous Maps

Let M be a compact manifold, with or without boundary. The genericity theorem of J. Palis, C. Pugh, M. Shub and D. Sullivan [PPSS] asserts that, among others, the property \( \Omega = \overline {\text{P}} \) (the set of non-wandering points is the closure of the set of periodic points) is C0-generic, i.e., holds for all homeomorphisms in some residual subset of the space Homeo(M) of all homeomorphisms of M to itself. This note points out and corrects a technical error in their proof, and extends the result to the space C0 (M, M) of all continuous maps of M to itself.

Pseudozeros of multivariate polynomials

The pseudozero set of a system f of polynomials in n complex variables is the subset of Cn which is the union of the zero-sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In particular it is shown that in many cases of natural interest, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets in coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials.

Real Algebraic Geometry and Ordered Structures

We show how one may sometimes perform singular ambient surgery on the complex locus of a real algebraic curve and obtain what we call a floppy curve. A floppy curve is a certain kind of singular surface in CP (2), more general than the complex locus of a real nodal curve. We derive analogs for floppy curves of known restrictions on real nodal curves. In particular we derive analogs of Fielder’s congruence for certain nonsingular curves and Viro’s inequalities for nodal curves which generalize those of Arnold and Petrovskii for nonsingular curves. We also obtain a determinant condition for certain curves which are extremal with respect to some of these equalities. One may prohibit certain schemes for real algebraic curves by prohibiting the floppy curves which result from singular ambient surgery. In this way, we give a new proof of Shustin’s prohibition of the scheme 1 < 2 ∐ 1 < 18 >> for a real algebraic curve of degree eight.

The H-Closed Monoreflections, Implicit Operations, and Countable Composition, in Archimedean Lattice-Ordered Groups with Weak Unit

In the category of the title, called W, we completely describe the monoreflections R

Comments on monoreflections in unital archimedean lattice-ordered groups and rings

\mathcal {R}

Monoreflections of commutative rings with identity

which are H-closed (closed under homomorphic image) by means of epimorphic extensions S of the free object on ω generators, F(ω), within the Baire functions on ℝω

Separating ideals in dimension 2

\mathbb {R}^{\omega }

“Mathematics Matters in Education” to Roger E. Howe and to All: An Introduction

, B(ℝω)

Knowing Ratio and Proportion for Teaching

B(\mathbb {R}^{\omega })