arXiv subjects

Sidney A. Morris

Publications and source records attributed to Sidney A. Morris.

15 recordsLinked to original sources

arXiv2026

Perfect Sets of Liouville Numbers with Controlled Self-Powers

We study the arithmetic behavior of self-powers $x^x$ when $x$ is a Liouville number. Using recent ideas on strengthened Liouville approximation, we develop flexible constructions that illuminate how transcendence, Liouville properties, and "large" topological size interact in this setting. As a concrete outcome, we build a perfect set of Liouville numbers of continuum cardinality whose finite sums, finite products, and self-powers all remain Liouville. These results show that rich algebraic and topological structures persist inside the Liouville universe for the map $x\mapsto x^x$.

math.NT↗

arXiv2024

Open mappings of locally compact groups

The aim of this note is to insert in the literature some easy but apparently not widely known facts about morphisms of locally compact groups, all of which are concerned with the openness of the morphism.

math.GR↗

arXiv2022

SIN, COS, EXP and LOG of Liouville numbers

For any Liouville number $α$, all of the following are transcendental numbers: $\textrm{e}^α$, $\log_\textrm{e}α$, $\sin α$, $\cosα$, $\tanα$, $\sinhα$, $\coshα$, $\tanhα$, $\arcsinα$ and the inverse functions evaluated at $α$ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if "Liouville number" is replaced by "$U$-number", where $U$ is one of Mahler's classes of transcendental numbers.

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arXiv2022

Topological Transcendental Fields

This article initiates the study of topological transcendental fields $\FF$ which are subfields of the topological field $\CC$ of all complex numbers such that $\FF$ consists of only rational numbers and a nonempty set of transcendental numbers. $\FF$, with the topology it inherits as a subspace of $\CC$, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is $\QQ(T)$, the extension of the field of rational numbers by a set $T$ of transcendental numbers. It is proved that there exist precisely $2^{\aleph_0}$ countably infinite topological transcendental fields and each is homeomorphic to the space $\QQ$ of rational numbers with its usual topology. It is also shown that there is a class of $2^{2^{\aleph_0} }$ of topological transcendental fields of the form $\QQ(T)$ with $T$ a set of Liouville numbers, no two of which are homeomorphic.

math.GN↗

arXiv2021

Transcendental Groups

In this note we introduce the notion of a transcendental group, that is, a subgroup $G$ of the topological group $\mathbb{C}$ of all complex numbers such that every element of $G$ except $ 0$ is a transcendental number. All such topological groups are separable metrizable zero-dimensional torsion-free abelian groups. Further, each transcendental group is homeomorphic to a subspace of $\mathbb{N}^{\aleph_0}$, where $\mathbb{N}$ denotes the discrete space of natural numbers. It is shown that (i) each countably infinite transcendental group is a member of one of three classes, where each class has $\mathfrak{c}$ (the cardinality of the continuum) members -- the first class consists of those isomorphic as a topological group to the discrete group $\ZZ$ of integers, the second class consists of those isomorphic as a topological group to $\ZZ\times \ZZ$, and the third class consists of those homeomorphic to the topological space $\QQ$ of all rational numbers; (ii) for each cardinal number $\aleph$ with $\aleph_0< \aleph\le \cc$, there exist $2^\aleph$ transcendental groups of cardinality $\aleph$ such that no two of the transcendental groups are isomorphic as topological groups or even homeomorphic; (iii) there exist $\mathfrak{c}$ countably infinite transcendental groups each of which is homeomorphic to $\QQ$ and algebraically isomorphic to a vector space over the field $\AAA$ of all algebraic numbers (and hence also over $\QQ$) of countably infinite dimension; (iv) $\RR$ has $2^\cc$ transcendental subgroups, each being a zero-dimensional metrizable torsion-free abelian group, such that no two of the transcendental groups are isomorphic as topological groups or even homeomorphic.

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arXiv2020

Tweaking Ramanujan's Approximation of n!

In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarino's argument allows a tweaking of Ramanujan's result to give a better approximation; (iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujan's result; (iv) that Chen's asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chen's.

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arXiv2018

A topological group observation on the Banach--Mazur separable quotient problem

The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fréchet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^ω$. It is proved in this paper that every infinite-dimensional Fréchet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace which is an infinite-dimensional Fréchet space, has an infinite-dimensional (in the topological sense) separable metrizable quotient group, namely $\mathbb{T}^ω$, where $\mathbb{T}$ denotes the compact unit circle group.

math.GN↗

arXiv2017

The Separable Quotient Problem for Topological Groups

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $σ$-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all $σ$-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.

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arXiv2016

Free topological vector spaces

We define and study the free topological vector space $\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\mathbb{V}(X)$ is a $k_ω$-space if and only if $X$ is a $k_ω$-space. If $X$ is infinite, then $\mathbb{V}(X)$ contains a closed vector subspace which is topologically isomorphic to $\mathbb{V}(\mathbb{N})$. It is proved that if $X$ is a $k$-space, then $\mathbb{V}(X)$ is locally convex if and only if $X$ is discrete and countable. If $X$ is a metrizable space it is shown that: (1) $\mathbb{V}(X)$ has countable tightness if and only if $X$ is separable, and (2) $\mathbb{V}(X)$ is a $k$-space if and only if $X$ is locally compact and separable. It is proved that $\mathbb{V}(X)$ is a barrelled topological vector space if and only if $X$ is discrete. This result is applied to free locally convex spaces $L(X)$ over a Tychonoff space $X$ by showing that: (1) $L(X)$ is quasibarrelled if and only if $L(X)$ is barrelled if and only if $X$ is discrete, and (2) $L(X)$ is a Baire space if and only if $X$ is finite.

math.GN↗

arXiv2015

Pro-Lie Groups: A survey with Open Problems

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally compact group which has a compact quotient group modulo its identity component and thus, in particular, each compact and each connected locally compact group; it also includes all locally compact abelian groups. This paper provides an overview of the structure theory and Lie theory of pro-Lie groups including results more recent than those in the authors' reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function which links the two. (A topological vector space is weakly complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of $\R$. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected with pro-Lie groups.

math.GR↗

arXiv2015

Density character of subgroups of topological groups

A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, locally compact abelian groups and connected locally compact groups and is closed under products and closed subgroups. A topological group G is almost connected if the quotient group of G by the connected component of its identity is compact. We prove that an almost connected pro-Lie group is separable iff its weight is not greater than c. It is deduced that an almost connected pro-Lie group is separable if and only if it is a subspace of a separable Hausdorff space. It is proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. Every precompact topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact group of weight c. This implies that there is a wealth of closed nonseparable subgroups of separable pseudocompact groups. An example is presented under CH of a separable countably compact abelian group which contains a non-separable closed subgroup. It is proved that the following conditions are equivalent for an omega-narrow topological group G: (i) G is a subspace of a separable regular space; (ii) G is a subgroup of a separable topological group; (iii) G is a closed subgroup of a separable pathconnected locally pathconnected group.

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arXiv2014

Nonmeasurable subgroups of compact groups

In 1985 S.~Saeki and K.~Stromberg published the following question: {\it Does every infinite compact group have a subgroup which is not Haar measurable?} An affirmative answer is given for all compact groups with the exception of some metric profinite groups known as strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup.

math.GR↗

arXiv2012

The weights of closed subgroups of a locally compact group

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le \lw(G)$, where $\lw(G)$ is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\aleph$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, $(τ_\aleph)_{\aleph_0\le \aleph\le \card(G)}$, such that $(G,τ_\aleph)\hat{\phantom{m}}\cong\hat G$. Items (2) and (3) are shown in Section 5.

math.GR↗

arXiv2011

Representing a profinite group as the homeomorphism group of a continuum

We contribute some information towards finding a general algorithm for constructing, for a given profinite group, $G$, a compact connected space, $X$, such that the full homeomorphism group, $H(X)$, with the compact-open topology is isomorphic to $G$ as a topological group. It is proposed that one should find a compact topological oriented graph $Γ$ such that $G\cong Aut(Γ)$. The replacement of the edges of $Γ$ by rigid continua should work as is exemplified in various instances where discrete graphs were used. It is shown here that the strategy can be implemented for profinite monothetic groups $G$.

math.GN↗

arXiv1999

Subgroups of monothetic groups

It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian group embeds into a metrizable monothetic group. More generally, we describe all topological groups that can be embedded into monothetic groups: they are exactly abelian topological groups of weight $\leq\frak c$ covered by countably many translations of every nonempty open subset.

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