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Anthony G. O'Farrell

Publications and source records attributed to Anthony G. O'Farrell.

At least 19 recordsLinked to original sources
arXiv2025

Forms of Nice Functions

You can invent striking and challenging problems with unique solution by building some symmetry into functional equations. Some are suitable for high school; others could generate college-level projects involving computer algebra. The problems are functional equations with group actions in the background. Interesting examples arise even from small finite groups. Whether a given problem ``works" with a given choice of constant coefficients depends on whether a related multilinear form is nonzero. These forms are essentially the classical group determinants studied by Frobenius in the nineteenth century.

math.GR↗
arXiv2025

Wiring switches to more light bulbs

Given $n$ buttons and $n$ bulbs so that the $i$th button toggles the $i$th bulb and perhaps some other bulbs, we compute the sharp lower bound on the number of bulbs that can be lit regardless of the action of the buttons. In the previous article we dealt with the case where each button affects at most 2 or 3 bulbs. In the present article we give sharp lower bounds for up to 4 or 5 wires per switch, and we show that the sharp asymptotic bound for an arbitrary number of wires is $\frac12$. (Even if you've found their buttons, you can please no more than half the people all the time!)

math.CO↗
arXiv2024

Wiring Switches to Light Bulbs

Given n buttons and n bulbs so that the ith button toggles the ith bulb and at most two other bulbs, we compute the sharp lower bound on the number of bulbs that can be lit regardless of the action of the buttons.

math.CO↗
arXiv2024

Functions with image in a strip

We consider holomorphic functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups.

math.FA↗
arXiv2023

Riordan Groups in higher Dimensions

The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix multiplication to two group actions on such series, namely formal (convolution) multiplication and formal composition. We define the analogous Riordan groups involving formal power series in several variables, and establish the analogue of the Fundamental Theorem in that context. We discuss related groups of Laurent series and pose some questions.

math.GR↗
arXiv2022

Dimension of Images of Large Level Sets

Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<α<1$ let $I_α(f)$ denote the set of values $y\in\mathbb{R}$ whose preimage $f^{-1}(y)$ has Hausdorff dimension $\dim f^{-1}(y) \ge α$. We consider how large can be the Hausdorff dimension of $I_α(f)$, as $f$ ranges over the set $C^k(E,\mathbb{R})$ of all $k$-times continuously-differentiable functions from $E$ into $\mathbb{R}$. We show that the sharp upper bound on $\dim I_α(f)$ is $\displaystyle\frac{1-α}k$.

math.CA↗
arXiv2022

Centralisers of formal maps

We consider formal maps in any finite dimension $d$ with coefficients in an integral domain $K$ with identity. Those invertible under formal composition form a group $\mathcal{G}$. We consider the centraliser $C_g$ of an element $g\in\mathcal{G}$ which is tangent to the identity of $\mathcal{G}$. Elements of finite order always have an uncountable centraliser. If $g$ has infinite order and $K$ is a field of characteristic zero we show that $C_g$ contains an isomorphic copy of the additive group $(K,+)$. If $g$ has infinite order and $K$ has finite characteristic we show that $C_g$ contains an uncountable abelian subgroup. The proofs are quite different in finite characteristic and in characteristic zero, but are connected by so-called sum functions.

math.GR↗
arXiv2022

Formally-Reversible Maps of C^2

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. This paper is about reversibles in the group $G$ of formally-invertible pairs of formal power series in two variables, with complex coefficients. The main result is a description of the generic reversible elements of $G$. We list two explicit sequences of reversibles which between them represent all the conjugacy classes of such reversibles. We show that each such element is reversible by some element of finite order, and hence is the product of two elements of finite even order. Those elements that may be reversed by an involution are called {\em strongly reversible}. We also characterise these. We draw some conclusions about generic reversibles in the group of biholomorphic germs in two variables, and about the factorization of formal maps as products of reversibles. Specifically, each product of reversibles reduces to the product of five.

math.CV↗
arXiv2019

Banach Algebras of Vector-valued Functions

We introduce the concept of an $E$-valued function algebra, a type of Banach algebra that consist of continuous $E$-valued functions on some compact Hausdorff space, where $E$ is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative $E$-valued function algebras. We give some specific examples.

math.FA↗
arXiv2019

Boundary values of holomorphic distributions in negative Lipschitz classes

We consider the behaviour at a boundary point of an open subset $U\subset\mathbb{C}$ of distributions that are holomorphic on $U$ and belong to what are called negative Lipschitz classes. The result explains the significance for holomorphic functions of series of Wiener type involving Hausdorff contents of dimension between $0$ and $1$. We begin with a survey about function spaces and capacities that sets the problem in context and reviews the relevant general theory. The techniques used include the construction of a special partition of the identity that may be of independent interest.

math.CV↗
arXiv2016

Derivatives at the Boundary for Analytic Lipschitz Functions

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $α$, with $0<α<1$, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point $b$, it may be evaluated by taking a limit of classical difference quotients, for approach from a set having full area density at $b$.

math.CV↗
arXiv2016

Ditkin Conditions

This paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra $E$ with unit and the associated commutative Banach algebra $C(X,E)$ of all continuous functions from a compact Hausdorff space $X$ into $E$. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by $E$ and $C(X,E)$. We also consider the relationship between these properties in the algebras $E$, $B$ and $\tilde B$ that appear in so-called admissible quadruples $(X,E,B,\tilde B)$.

math.FA↗
arXiv2015

Boundary smoothness of analytic functions

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $α$, with $0<α<1$, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We obtain a result which applies, for example, when the open set admits an interior cone at the special boundary point.

math.CV↗
arXiv2014

Geometry in the Transition from Primary to Post-Primary

This article is intended as a kind of precursor to the document Geometry for Post-primary School Mathematics, part of the Mathematics Syllabus for Junior Certicate issued by the Irish National Council for Curriculum and Assessment in the context of Project Maths. Our purpose is to place that document in the context of an overview of plane geometry, touching on several important pedagogical and historical aspects, in the hope that this will prove useful for teachers.

math.HO↗
arXiv2013

Factoring Formal Maps into Reversible or Involutive Factors

An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group $\mathfrak{G}_n$ of formal maps of $(\mathbb{C}^n,0)$, i.e. formally-invertible $n$-tuples of formal power series in $n$ variables, with complex coefficients. The case $n=1$ was already understood. Each product $F$ of reversibles has linear part $L(F)$ of determinant $\pm1$. The main results are that for $n\ge2$ each map $F$ with det$(L(F))=\pm1$ is the product of $2+3c$ reversibles, and may also be factored as the product of $9+6c$ involutions, where $c$ is the smallest integer $\ge \log_2n$.

math.GR↗
arXiv2013

Reversibility and Banach Algebras

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. In this paper we review some results about the structure of groups involving the reversible elements and we pose some questions about groups associated to a Banach algebra.

math.GR↗
arXiv2011

Conjugacy of real diffeomorphisms. A survey

Given a group G, the conjugacy problem in G is the problem of giving an effective procedure for determining whether or not two given elements f, g of G are conjugate, i.e. whether there exists h belonging to G with fh = hg. This paper is about the conjugacy problem in the group Diffeo(I) of all diffeomorphisms of an interval I in R. There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.

math.DS↗