Tom Richmond

How to Recognize a Parabola

Parabolas have many interesting properties which were perhaps more well known in centuries past. Many of these properties hold only for parabolas, providing a characterization which can be used to recognize (theoretically, at least) a parabola. Here, we present a dozen characterizations of parabolas, involving tangent lines, areas, and the well-known reflective property. While some of these properties are widely known to hold for parabolas, the fact that they hold only for parabolas may be less well known. These remarkable properties can be verified using only elementary techniques of calculus, geometry, and differential equations. A parabola is the set of points in the plane which are equidistant from a point F called the focus and a line / called the directrix. If the directrix is horizontal, then the parabola is the graph of a quadratic function p(x) = ax2 + sx + y. We will not distinguish between a function and its graph. A chord of a function f(x) is a line segment whose endpoints lie on the graph of the function. A chord of f(x) is a segment of a secant line to f(x). By the equation of a chord, we mean the equation of the corresponding secant line. Our first characterization of parabolas involves the area between a function and a chord having horizontal extent h.

Quasiorders, principal topologies, and partially ordered partitions

The quasiorders on a setX are equivalent to the topologies onX which are closed under arbitrary intersections. We consider the quaisorders onX to be partial orders on the blocks of a partition of X and use this approach to survey some fundamental results on the lattice of quasiorders on X.

The equal area zones property

A classical exercise found in many calculus texts is the zones of a sphere problem: Verify that if a sphere is sliced by two parallel planes h units apart, then the surface area of the zone between the planes is dependent on h alone, and independent of the location of the planes. While this property of the sphere is usually surprising to students, it has long been known. It is an immediate consequence of the following result of Archimedes (see p. 185ff in [1]) illustrated by the figure below.

The Lattice of Ordered Compactifications of a Direct Sum of Totally Ordered Spaces

The lattice of ordered compactifications of a topological sum of a finite number of totally ordered spaces is investigated. This investigation proceeds by decomposing the lattice into equivalence classes determined by the identification of essential pairs of singularities. This lattice of equivalence classes is isomorphic to a power set lattice. Each of these equivalence classes is further decomposed into equivalence classes determined by admissible partially ordered partitions of the ordered Stone–Čech remainder. The lattice structure within each equivalence class is determined using an algorithm based on the incidence matrix of the partially ordered partition. As examples, the ordered compactification lattices for the spaces [0,1)⊕[0,1),[0,1)⊕[0,1)⊕[0,1),R⊕R, and R/{0}⊕R/{0} are determined.

On the Nachbin compactification of products of totally ordered spaces

Necessary and sufficient conditions are given for β 0 ( X × Y ) = β 0 X × β 0 Y , where X and Y are totally ordered spaces and β 0 X denotes the Nachbin (or Stone-Cech ordered) compactification of X .

Cardinality and Structure of Semilattices of Ordered Compactifications

ABSTRACT: Cardinalities and lattice structures which are attainable by semilattices of ordered compactifications of completely regular ordered spaces are examined. Visliseni and Flachsmeyer have shown that every infinite cardinal is attainable as the cardinality of a semilattice of compactifications of a Tychonoff space. Among the finite cardinals, however, only the Bell numbers are attainable as cardinalities of semilattices of compactifications. It is shown here that all cardinals, both finite and infinite, are attainable as the cardinalities of semilattices of ordered compactifications of completely regular ordered spaces. The last section examines lattice structures which are realizable as semilattices of ordered compactifications, such as chains and power sets.

Completely regularly ordered spaces versus T2-ordered spaces which are completely regular☆

Schwarz and Weck-Schwarz have shown that a T2-ordered space (X, τ,) whose underlying topological space (X, τ ) is completely regular need not be a completely regularly ordered space (that is, T3.5 + T2-orderedT3.5-ordered). We show that a completely regular T2-ordered space need not be completely regularly ordered even under more stringent assumptions such as convexity of the topology. One example involves the construction of a nontrivial topological ordered space on which every continuous increasing function into the real unit interval is constant.  2003 Elsevier B.V. All rights reserved. MSC: 54F05; 54G20; 54C99; 06F30

Separation properties of the Wallman ordered compactification

The Wallman ordered compactification ω0X of a topological ordered space X is T2-ordered (and hence equivalent to the Stone-Cech ordered compactification) iff X is a T4-ordered c-space. In particular, these two ordered compactifications are equivalent when X is n dimensional Euclidean space iff n≤2. When X is a c-space, ω0X is T1-ordered; we give conditions on X under which the converse statement is also true. We also find conditions on X which are necessary and sufficient for ω0X to be T2. Several examples provide further insight into the separation properties of ω0X.

Order-theoretical connectivity

Order-theoretically connected posets are introduced and applied to create the notion of T-connectivity in ordered topological spaces. As special cases T-connectivity contains classical connectivity, order-connectivity, and link-connectivity.

Finite-point order compactifications

After the characterization of 1-point topological compactifications by Alexandroff in 1924, n -point topological compactifications by Magill [4] in 1965, and 1-point order compactifications by McCallion [5] in 1971, spaces that admit an n -point order compactification are characterized in Section 2. If X * and X ** are finite-point order compactifications of X , sup{ X *, X **} is given explicitly in terms of X * and X ** in § 3. In § 4 it is shown that if an ordered topological space X has an m -point and an n -point order compactification, then X has a k -point order compactification for each integer k between m and n. The author is indebted to Professor Darrell C. Kent, who provided assistance and encouragement during the preparation of this paper.