Local connectivity functions on arcwise connected spaces and certain continua

Types of spaces are given on which every local connectivity function is a connectivity function, a connected function, or a Darboux function. A complete determination such spaces is obtained when the spaces are assumed to be arc-like continua or circle-like continua. Results provide answers to a question asked by Stallings.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.     

Sums of Connectivity Functions on Rn

A function f: Rⁿ R is a connectivity function if the graphof its restriction f|C to any connected C Rⁿ is connected inRⁿ x R. The main goal of this paper is to prove that every functionf: Rⁿ R is a sum of n + 1 connectivity functions (Corollary2.2). We will also show that if n > 1, then every functiong: Rⁿ R which is a sum of n connectivity functions is continuouson some perfect set (see Theorem 2.5) which implies that thenumber n + 1 in our theorem is best possible (Corollary 2.6). Toprove the above results, we establish and then apply the followingtheorems which are of interest on their own. For every dense G-subset G of Rⁿ there are homeomorphisms h₁,..., h_n of Rⁿ such that Rⁿ = G h₁(G) ... h_n(G) (Proposition2.4). For every n > 1 and any connectivity function f: Rⁿ R, ifx Rⁿ and > 0 then there exists an open set U Rⁿ such thatx U Bⁿ(x, ), f|bd(U) is continuous, and |(x) – f(y)|< for every y bd(U) (Proposition 2.7). 1991 MathematicsSubject Classification: 26B40, 54C30, 54F45.     

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace Cn(X), n<∞, is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of Cn(X) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for Cn(X) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C^*-smoothness. Some new results are obtained for C(X), the space of subcontinua of X. A class of continua X is given for which Cn(X) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.     

The nonexistence of almost continuous surjections between certain continua

Theorems about the nonexistence of continuous surjections between continua and related results are extended to almost continuous surjections. Several questions are posed.     

On the Length of Snakes in Powers of Complete Graphs

The conjecture stated in an earlier paper by the author thatthere is a constant (independent from both n and k) such that n^d–1 holds for everyn 2 and d 2, where is the length of the longest snake (cycle without chords) in the Cartesianproduct of d copies of thecomplete graph K_n, is proved.