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Let
$P(n)$
denote the largest prime factor of an integer
$n\geq 2$
. In this paper, we study the distribution of the sequence
$\{f(P(n)):n\geq 1\}$
over the set of congruence classes modulo an integer
$b\geq 2$
, where
$f$
is a strongly
$q$
-additive integer-valued function (that is,
$f(aq^{j}+b)=f(a)+f(b),$
with
$(a,b,j)\in \mathbb{N}^{3}$
,
$0\leq b). We also show that the sequence
$\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$
is uniformly distributed modulo 1 if and only if
${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$
.
We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.
We prove an asymptotic formula for the sum
$\sum _{n\leq N}d(n^{2}-1)$
, where
$d(n)$
denotes the number of divisors of
$n$
. During the course of our proof, we also furnish an asymptotic formula for the sum
$\sum _{d\leq N}g(d)$
, where
$g(d)$
denotes the number of solutions
$x$
in
$\mathbb{Z}_{d}$
to the equation
$x^{2}\equiv 1~(\text{mod}~d)$
.
Let
$K$
be a number field with ring of integers
${\mathcal{O}}$
. After introducing a suitable notion of density for subsets of
${\mathcal{O}}$
, generalising the natural density for subsets of
$\mathbb{Z}$
, we show that the density of the set of coprime
$m$
-tuples of algebraic integers is
$1/{\it\zeta}_{K}(m)$
, where
${\it\zeta}_{K}$
is the Dedekind zeta function of
$K$
. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math.77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis3 (1883), 224–225] concerning the density of coprime pairs of integers in
$\mathbb{Z}$
.
Let
$m$
be a positive integer and
$p$
a prime number. We prove the orthogonality of some character sums over the finite field
$\mathbb{F}_{p^{m}}$
or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension
$p^{m}$
over the complex number field
$\mathbb{C}$
, especially with
$p=2$
.
Using the inductive structure of a Fermat variety by Shioda and Katsura [‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115], we estimate the refined motivic dimension of certain Fermat varieties. As an application of our computation, we present an elementary proof of the generalised Hodge conjecture for those varieties.
In this paper we prove the following result: let
$m,n\geq 1$
be distinct integers, let
$R$
be an
$mn(m+n)|m-n|$
-torsion free semiprime ring and let
$D:R\rightarrow R$
be an
$(m,n)$
-Jordan derivation, that is an additive mapping satisfying the relation
$(m+n)D(x^{2})=2mD(x)x+2nxD(x)$
for
$x\in R$
. Then
$D$
is a derivation which maps
$R$
into its centre.
Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has
$\text{A}_{5}$
as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of
$4$
-power-free order.
The Brück conjecture states that if a nonconstant entire function
$f$
with hyper-order
${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$
shares one finite value
$a$
(counting multiplicities) with its derivative
$f^{\prime }$
, then
$f^{\prime }-a=c(f-a)$
, where
$c$
is a nonzero constant. The conjecture has been established for entire functions with order
${\it\sigma}(f)<+\infty$
and hyper-order
${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$
. The purpose of this paper is to prove the Brück conjecture for the case
${\it\sigma}_{2}(f)=\frac{1}{2}$
by studying the infinite hyper-order solutions of the linear differential equations
$f^{(k)}+A(z)f=Q(z)$
. The shared value
$a$
is extended to be a ‘small’ function with respect to the entire function
$f$
.
We obtain lower bounds for the growth of solutions of higher order linear differential equations, with coefficients analytic in the unit disc of the complex plane, by localising the equations via conformal maps and applying known results for the unit disc. As an example, we study equations in which the coefficients have a certain explicit exponential growth at one point on the boundary of the unit disc and consider the iterated
$M$
-order of solutions.
Let
$G$
be a commutative group,
$Y$
a real Banach space and
$f:G\rightarrow Y$
. We prove the Ulam–Hyers stability theorem for the cyclic functional equation
$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$
for all
$x,y\in {\rm\Omega}$
, where
$H$
is a finite cyclic subgroup of
$\text{Aut}(G)$
and
${\rm\Omega}\subset G\times G$
satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation
$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$
for all
$(z,{\it\zeta})\in {\rm\Omega}$
, where
$f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$
and
${\rm\Omega}\subset \mathbb{C}^{2}$
has four-dimensional Lebesgue measure
$0$
.
Given a closed set
$C$
in a Banach space
$(X,\Vert \cdot \Vert )$
, a point
$x\in X$
is said to have a nearest point in
$C$
if there exists
$z\in C$
such that
$d_{C}(x)=\Vert x-z\Vert$
, where
$d_{C}$
is the distance of
$x$
from
$C$
. We survey the problem of studying the size of the set of points in
$X$
which have nearest points in
$C$
. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.
From the viewpoint of
$C^{\ast }$
-dynamical systems, we define a weak version of the Haagerup property for the group action on a
$C^{\ast }$
-algebra. We prove that this group action preserves the Haagerup property of
$C^{\ast }$
-algebras in the sense of Dong [‘Haagerup property for
$C^{\ast }$
-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product
$C^{\ast }$
-algebra
$A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$
has the Haagerup property with respect to the induced faithful tracial state
$\widetilde{{\it\tau}}$
if
$A$
has the Haagerup property with respect to
${\it\tau}$
.
We show that, under special hypotheses, each 3-Jordan homomorphism
${\it\varphi}$
between Banach algebras
${\mathcal{A}}$
and
${\mathcal{B}}$
is a 3-homomorphism.
We examine a class of ergodic transformations on a probability measure space
$(X,{\it\mu})$
and show that they extend to representations of
${\mathcal{B}}(L^{2}(X,{\it\mu}))$
that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of
$X$
for
$N$
-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.
In this paper, we establish a convergence theorem for fixed points of generalised weak contractions in complete metric spaces under some new control conditions on the functions. An illustrative example of a generalised weak contraction is discussed to show how the new conditions extend known results.
We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.