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It is proved that the free topological vector space
$\mathbb{V}([0,1])$
contains an isomorphic copy of the free topological vector space
$\mathbb{V}([0,1]^{n})$
for every finite-dimensional cube
$[0,1]^{n}$
, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval
$[0,1]$
to general metrisable spaces. Indeed, we prove that the free topological vector space
$\mathbb{V}(X)$
does not even have a vector subspace isomorphic as a topological vector space to
$\mathbb{V}(X\oplus X)$
, where
$X$
is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.
Let
$\mathbf{H}_{\mathbb{H}}^{n}$
denote the
$n$
-dimensional quaternionic hyperbolic space. The linear group
$\text{Sp}(n,1)$
acts on
$\mathbf{H}_{\mathbb{H}}^{n}$
by isometries. A subgroup
$G$
of
$\text{Sp}(n,1)$
is called Zariski dense if it neither fixes a point on
$\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$
nor preserves a totally geodesic subspace of
$\mathbf{H}_{\mathbb{H}}^{n}$
. We prove that a Zariski dense subgroup
$G$
of
$\text{Sp}(n,1)$
is discrete if for every loxodromic element
$g\in G$
the two-generator subgroup
$\langle f,gfg^{-1}\rangle$
is discrete, where the generator
$f\in \text{Sp}(n,1)$
is a certain fixed element not necessarily from
$G$
.
We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.
We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation
$f(x+y)=f(x)+f(y)$
, but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a problem of Th. M. Rassias. The main tool is a recent fixed point theorem in some spaces of functions with values in 2-Banach spaces.
In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.
An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.
We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph
$K_{16}$
. Of these, 89 have a nontrivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [‘Perfect 1-factorisations of
$K_{16}$
with nontrivial automorphism group’, J. Combin. Math. Combin. Comput.47 (2003), 97–111]). We also (i) describe a new invariant which distinguishes between the P1Fs of
$K_{16}$
, (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.
In the field
$\mathbb{K}$
of formal power series over a finite field
$K$
, we consider some lacunary power series with algebraic coefficients in a finite extension of
$K(x)$
. We show that the values of these series at nonzero algebraic arguments in
$\mathbb{K}$
are
$U$
-numbers.
We generalise a result of Chern [‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math.47 (2017), 23–26] on distinct partitions with bounded difference between largest and smallest parts. The generalisation is proved both analytically and bijectively.
We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.
Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form.
Given a positive integer
$m$
, a finite
$p$
-group
$G$
is called a
$BC(p^{m})$
-group if
$|H_{G}|\leq p^{m}$
for every nonnormal subgroup
$H$
of
$G$
, where
$H_{G}$
is the normal core of
$H$
in
$G$
. We show that
$m+2$
is an upper bound for the nilpotent class of a finite
$BC(p^{m})$
-group and obtain a necessary and sufficient condition for a
$p$
-group to be of maximal class. We also classify the
$BC(p)$
-groups.
This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if
$A$
and
$B$
are nontrivial free subgroups of a virtually free group containing a free subgroup of index
$n$
, then
$\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$
. In addition, we obtain a virtually-free-product analogue of this result.
We prove the global logarithmic stability of the Cauchy problem for
$H^{2}$
-solutions of an anisotropic elliptic equation in a Lipschitz domain. The result is based on existing techniques used to establish stability estimates for the Cauchy problem combined with related tools used to study an inverse medium problem.
which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author [‘A
$q$
-analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl.466 (2018), 749–761].
Let
$n$
be a positive integer and
$a$
an integer prime to
$n$
. Multiplication by
$a$
induces a permutation over
$\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$
. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo
$p$
and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary
$k$
th power residues modulo
$p$
and primitive roots modulo a power of
$p$
.
A polynomial
$f$
over a finite field
$\mathbb{F}_{q}$
can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers
$f^{e}$
for each
$e$
from
$1$
to
$q-2$
, as well as the existence of a unique solution to
$f(x)=0$
in
$\mathbb{F}_{q}$
. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.
Let
$\mathfrak{F}$
be a class of finite groups and
$G$
a finite group. Let
${\mathcal{L}}_{\mathfrak{F}}(G)$
be the set of all subgroups
$A$
of
$G$
with
$A^{G}/A_{G}\in \mathfrak{F}$
. A chief factor
$H/K$
of
$G$
is
$\mathfrak{F}$
-central in
$G$
if
$(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$
. We study the structure of
$G$
under the hypothesis that every chief factor of
$G$
between
$A_{G}$
and
$A^{G}$
is
$\mathfrak{F}$
-central in
$G$
for every subgroup
$A\in {\mathcal{L}}_{\mathfrak{F}}(G)$
. As an application, we prove that a finite soluble group
$G$
is a PST-group if and only if
$A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$
for every subgroup
$A\in {\mathcal{L}}_{\mathfrak{N}}(G)$
, where
$\mathfrak{N}$
is the class of all nilpotent groups.
We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.