arXiv subjects

Paul Gartside

Publications and source records attributed to Paul Gartside.

24 records · Page 2Linked to original sources

arXiv2012

Strong Arcwise Connectedness

A space is `n-strong arc connected' (n-sac) if for any n points in the space there is an arc in the space visiting them in order. A space is omega-strong arc connected (omega-sac) if it is n-sac for all n. We study these properties in finite graphs, regular continua, and rational continua. There are no 4-sac graphs, but there are 3-sac graphs and graphs which are 2-sac but not 3-sac. For every n there is an n-sac regular continuum, but no regular continuum is omega-sac. There is an omega-sac rational continuum. For graphs we give a simple characterization of those graphs which are 3-sac. It is shown, using ideas from descriptive set theory, that there is no simple characterization of n-sac, or omega-sac, rational continua.

math.GN↗

arXiv2009

On Hilbert's 13th Problem

Every continuous function of two or more real variables can be written as the superposition of continuous functions of one real variable along with addition.

math.CA↗

arXiv2009

Minimal Size of Basic Families

A family $\bfam$ of continuous real-valued functions on a space $X$ is said to be {\sl basic} if every $f \in C(X)$ can be represented $f = \sum_{i=1}^n g_i \circ ϕ_i$ for some $ϕ_i \in \bfam$ and $g_i \in C(\R)$ ($i=1, ..., n$). Define $\basic (X) = \min \{|\bfam| : \bfam$ is a basic family for $X\}$. If $X$ is separable metrizable $X$ then either $X$ is locally compact and finite dimensional, and $\basic (X) < \aleph_0$, or $\basic (X) = \mathfrak{c}$. If $K$ is compact and either $w(K)$ (the minimal size of a basis for $K$) has uncountable cofinality or $K$ has a discrete subset $D$ with $|D|=w(K)$ then either $K$ is finite dimensional, and $\basic (K) = \cof ([w(K)]^{\aleph_0}, \subseteq)$, or $\basic (K) = |C(K)|=w(K)^{\aleph_0}$.

math.GN↗

arXiv2008

Counting the Closed Subgroups of Profinite Groups

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.

math.GR↗

arXiv2008

Classifying Spaces of Subgroups of Profinite Groups

The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered height.

math.GR↗

arXiv2008

Spaces with a Finite Family of Basic Functions

A space X is finite dimensional, locally compact and separable metrizable if and only if X has a finite basic family: continuous functions Phi_1,...,Phi_n of X to the reals, R, such that for all continuous f from X to R there are g_1,..., g_n in C(R) satisfying f(x)=g_1(Phi_1(x))+g_2(Phi_2(x))+...+g_n(Phi_n(x)) for all x in X. This give the complete solution to four problems on basic functions posed by Sternfeld, as well as questions posed by Hattori and others.

math.FA↗