text:
\begin{document}
\title{Remarks on Fixed Point Assertions in Digital Topology}
\author{Laurence Boxer
         \thanks{
    Department of Computer and Information Sciences,
    Niagara University,
    Niagara University, NY 14109, USA;
    and Department of Computer Science and Engineering,
    State University of New York at Buffalo.
    E-mail: boxer@niagara.edu
    }
    \and{
    P. Christopher Staecker
    \thanks{Department of Mathematics, Fairfield University, Fairfield, CT 06823-5195, USA.
    E-mail: cstaecker@fairfield.edu
    }
    }
}
\date{ }
\maketitle

\begin{abstract}
Several recent papers in digital topology have sought to
obtain fixed point results by mimicking the use of tools
from classical topology, such as complete metric spaces
and homotopy invariant fixed point theory. We show that
in many cases, researchers using these tools have derived
conclusions that are incorrect or trivial.

Key words and phrases: digital topology, fixed point,
metric space
\end{abstract}

\section{Introduction}
Recent papers have attempted to apply to digital images
ideas from Euclidean topology and real analysis concerning
metrics and fixed points. While the underlying motivation
of digital topology comes from Euclidean topology and real
analysis, some applications recently featured in the
literature seem of doubtful worth. Although papers
including~\cite{Rosenfeld87,Bx-etal} have valid and
interesting results for fixed points and for ``almost" or
``approximate" fixed points in digital topology,
many other published assertions concerning fixed points in
digital topology are incorrect or trivial (e.g., applicable
only to singletons, or functions studied forced to be
constant), as we will discuss in the current paper.

\section{Preliminaries}
We let $\Z$ denote the set of integers, and $\R$, the real line.

We consider a digital image as a graph $(X,\kappa)$, where
$X \subset \Z^n$ for some positive integer $n$ and
$\kappa$ is an adjacency relation on $X$.

A digital metric space is~\cite{EgeKaraca} a triple
$(X,d,\kappa)$ where $(X,\kappa)$ is a digital image and
$d$ is a metric for $X$. In~\cite{EgeKaraca}, $d$ was taken
to be the Euclidean metric, but we will not limit our
discussion to the Euclidean metric.

The {\em diameter} of a metric space $(X,d)$
is
\[diam \, X = \max \{d(x,y) \, | \, x,y \in X\}.\]

\subsection{Adjacencies}
The most commonly used adjacencies for digital images
are the $c_u$-adjacencies, defined as follows.
\begin{definition}
Let $p,q \in \Z^n$, $p=(p_1,\ldots,p_n)$,
$q=(q_1,\ldots,q_n)$, $p \ne q$. Let $1 \le u \le n$. We say
$p$ and $q$ are $c_u$-adjacent, denoted
$p \adj_{c_u} q$ or $p \adj q$ when the adjacency is
understood, if
\begin{itemize}
    \item for at most $u$ distinct indices $i$,
          $|p_i - q_i| = 1$, and
    \item for all other indices $j$, $p_j = q_j$.
\end{itemize}
\end{definition}

Often, a $c_u$-adjacency is denoted by the number
of points in $\Z^n$ that are $c_u$-adjacent to a given
point. E.g.,
\begin{itemize}
    \item in $\Z^1$, $c_1$-adjacency is 2-adjacency;
    \item in $\Z^2$, $c_1$-adjacency is 4-adjacency and
          $c_2$-adjacency is 8-adjacency;
    \item in $\Z^3$, $c_1$-adjacency is 8-adjacency,
          $c_2$-adjacency is 18-adjacency, and
          $c_3$-adjacency is 26-adjacency.
\end{itemize}

An adjacency often used for Cartesian products of
digital images is the
{\em normal product adjacency}, denoted in the following by
$\kappa_*$ and defined~\cite{Berge}
as follows. Given digital images $(X,\kappa)$ and
$(Y,\lambda)$ and points $x, x' \in X$, $y,y' \in Y$,
we have $(x,y) \adj_{\kappa_*} (x',y')$ in $X \times Y$
if and only if one of the following holds.
\begin{itemize}
    \item $x \adj_{\kappa} x'$ and $y=y'$, or
    \item $x = x'$ and $y \adj_{\lambda} y'$, or
    \item $x \adj_{\kappa} x'$ and $y \adj_{\lambda} y'$.
\end{itemize}

Other adjacencies for digital images are discussed in
papers such as \cite{Herman,Bx17,Bx18}.

A {\em digital interval} is a digital image of the form
$([a,b]_Z, 2)$, where $a < b$ and
$[a,b]_Z = \{z \in \Z \, | \, a \le z \le b\}$.

\subsection{$\ell_p$ metric}
Let $X \subset \R^n$ and let $x=(x_1,\ldots, x_n)$ and
$y=(y_1,\ldots,y_n)$ be points of $X$. Let
$1 \le p \le \infty$. The $\ell_p$ metric $d$ for $X$ is defined by
\[ d(x,y) = \left \{ \begin{array}{ll}
        \left (    \sum_{i=1}^n |x_i - y_i|^p \right )^{1/p} &
            \mbox{for } 1 \le p < \infty; \\
            \max \{|x_i - y_i|\}_{i=1}^n &
            \mbox{for } p = \infty. \end{array}
\right .
\]
For $p=1$, this gives us the {\em Manhattan metric}
$d(x,y) = \sum_{i=1}^n |x_i - y_i|$; for $p=2$, we have the
{\em Euclidean metric} $d(x,y) = (\sum_{i=1}^n |x_i - y_i|^2)^{1/2}$.

Notice that for any $\ell_p$ metric $d$, if $x,y \in \Z^n$ and
$d(x,y) < 1$, then $x=y$.

\subsection{Digital continuity and homotopy}
\begin{definition}
\label{digitalContinuity}
\cite{Rosenfeld87,Bx99}
A function $f: (X,\kappa) \to (Y, \lambda)$ between
digital images is $(\kappa,\lambda)$-{\em continuous}
(or just {\em continuous} when $\kappa$ and $\lambda$ are
understood) if for every $\kappa$-connected subset
$X'$ of $X$, $f(X')$ is a $\lambda$-connected subset of $Y$.
\end{definition}

\begin{thm}
{\rm \cite{Bx99}}
A function $f: (X,\kappa) \to (Y, \lambda)$ between
digital images is $(\kappa,\lambda)$-{\em continuous} if
and only if $x \adj_{\kappa} x'$ in $X$ implies either
$f(x)=f(x')$ or $f(x) \adj_{\lambda} f(x')$ in $Y$.
\end{thm}

As in topology, the digital topology notion of homotopy
can be understood as one function deforming in a continuous
fashion into another. Precisely, we have the following.

\begin{definition}
\cite{Bx99}
Let $f,g: (X,\kappa) \to (Y,\lambda)$. We say
$f$ and $g$ are homotopic, denoted
$f \simeq_{(\kappa,\lambda)} g$ or $f \simeq g$ when
$\kappa$ and $\lambda$ are understood, if there is
a function $F: X \times [0,m]_{\Z} \to Y$ for some
$m \in \N$ such that
\begin{itemize}
    \item $F(x,0)=f(x)$ and $F(x,m)=g(x)$ for all $x \in X$.
    \item The induced function $F_t: X \to Y$ defined by
          $F_t(x)=F(x,t)$ is $(\kappa,\lambda)$-continuous
          for all $t \in [0,m]_{\Z}$.
    \item The induced function $F_x: [0,m]_{\Z} \to Y$
          defined by $F_x(t)=F(x,t)$ is
          $(2,\lambda)$-continuous for all $x \in X$.
\end{itemize}
\end{definition}

\subsection{Cauchy sequences and complete metric spaces}
The papers~\cite{EgeKaraca,Han16,Hossain-etal,Jain,JR17,JR18,Mishra-etal,Ege-etal} apply to
digital images the notions of Cauchy
sequence and complete metric space.
Since for common metrics such as an $\ell_p$ metric, a digital metric space is discrete, the digital versions of
these notions are quite limited.

Recall that a sequence of points $\{x_n\}$ in a metric space
$(X,d)$ is a {\em Cauchy sequence}
if for all $\varepsilon > 0$ there exists $n_0 \in \N$
such that $m,n > n_0$ implies
$d(x_m,x_n) < \varepsilon$. If every Cauchy sequence in $X$ has a limit, then $(X,d)$ is a
{\em complete metric space}.

It has been shown that under a mild additional assumption, a digital
Cauchy sequence is eventually constant. The following is an easy
generalization of Proposition~3.6 of~\cite{Han16}, where only
the Euclidean metric was considered. The proof given in~\cite{Han16}
is easily modified to give the following.

\begin{thm}
\label{Han-Cauchy}
Let $a > 0$. If $d$ is a metric on a digital image $(X,\kappa)$ such that
for all distinct $x,y \in X$ we have $d(x,y) > a$, then
for any Cauchy sequence $\{x_i\}_{i=1}^{\infty} \subset X$
there exists $n_0 \in \N$ such that $m,n > n_0$ implies $x_m = x_n$.
\end{thm}

An immediate consequence of Theorem~\ref{Han-Cauchy} is the
following.

\begin{cor}
{\rm \cite{Han16}}
\label{Han-Cauchy-cor}
Let $(X,d,\kappa)$ be a digital metric space. If $d$ is a metric on $(X,\kappa)$ such
that for all distinct $x,y \in X$ we have $d(x,y) > a$ for some constant $a > 0$,
then any Cauchy sequence in $X$ is eventually constant, and $(X,d)$ is a complete metric space.
\end{cor}

\begin{remark}
{\rm
It is easily seen that the hypotheses of
Theorem~\ref{Han-Cauchy} and
Corollary~\ref{Han-Cauchy-cor}
are satisfied for any finite digital metric space, or for a
digital metric space $(X,d,\kappa)$ for which the metric $d$
is any $\ell_p$ metric. Thus, a Cauchy sequence that is
not eventually constant can only occur in an infinite
digital metric space with an unusual metric. Such an
example is given below.
}
\end{remark}

\begin{exl}
Let $d$ be the metric on $(\N,c_1)$ defined by
$d(i,j) = |1/i - 1/j|$. Then $\{i\}_{i=1}^{\infty}$ is
a Cauchy sequence for this metric that does not have a
limit.
\end{exl}

\subsection{Digital fixed points and approximate fixed points}
The study of the fixed points of continuous self-maps
is prominent in many areas of mathematics. We say
a topological space or a digital image $X$ has the
{\em fixed point property} if every continuous (respectively,
digitally continuous)
$f: X \to X$ has a fixed point, i.e., a point $p \in X$ such
that $f(p)=p$.

A version of Theorem~\ref{singleton} below was proved by Rosenfeld in \cite{Rosenfeld87} for the case when $X$ is a \emph{digital picture}, that is, a digital image of the form
$\Pi_{i=1}^n [a_i,b_i]_{\Z} \subset \Z^n$ with the
$c_n$-adjacency. For general digital images,
Theorem \ref{singleton} was proved in \cite{Bx-etal}.

\begin{thm}
\label{singleton}
A digital image $(X,\kappa)$ has the fixed point property
if and only if $X$ is a singleton.
\end{thm}

This theorem led to the study in~\cite{Bx-etal}
of the {\em approximate fixed point property}, an idea
suggested by results of~\cite{Rosenfeld87}. An
{\em approximate fixed point}~\cite{Bx-etal}, called an
{\em almost fixed point} in~\cite{Rosenfeld87}, of a
continuous function $f: (X,\kappa) \to (X,\kappa)$ is a point $p \in X$
such that $f(p)=p$ or $f(p) \adj_{\kappa} p$.
A digital image $(X,\kappa)$ has the {\em approximate
fixed point property (AFPP)}~\cite{Bx-etal} if for
every continuous $f: X \to X$ there is an approximate
fixed point of $f$.

We have rephrased the following to conform
with terminology used in this paper.

\begin{thm}
{\rm (Theorem~4.1 of~{\rm
\cite{Rosenfeld87}})}
\label{4.1Rose}
Every digital picture
$(\Pi_{i=1}^n [a_i,b_i]_{\Z}, c_n)$ has the AFPP.
\end{thm}

In Remark 6.2 (2) of~\cite{Han17}, the author
incorrectly attributes to~\cite{Rosenfeld87} the claim that
``Every digital image $(Y, 8)$ has the AFPP." The
attribution is incorrect, since, as we have shown above,
the citation should be about digital pictures, not the
more general digital images. Further, the claim is false, as the following example shows.

\begin{exl}
Let $n \ge 4$ and let
$Y = (\{y_i\}_{i=0}^{n-1},8) \subset \Z^2$ be a digital
simple closed curve with the points $y_i$ indexed
circularly. Then $(Y,8)$ does not have the AFPP.
\end{exl}

\begin{proof}
The function $f: Y \to Y$ defined by
$f(y_i) = y_{(i+2) \mod n}$ is easily seen to be
$(8,8)$-continuous and free of approximate fixed points.
\end{proof}

\subsection{Contraction and expansion functions}
\label{contractionFunctions}
We introduce several fixed point theorems for digital topology that
are modeled on analogs for the topology and analysis of $\R^n$.

In the following definitions, we assume $(X,d,\kappa)$ is a digital
metric space and $f: X \to X$ is a function.

\begin{definition}
\label{contractionDef}
{\rm \cite{EgeKaraca}}
If for some $\alpha \in (0,1)$ and all
          $x,y \in X$, $d(f(x),f(y)) < \alpha d(x,y)$, then
          $f$ is a {\em digital contraction map}. We say
$\alpha$ is the {\em multiplier}.
\end{definition}

Note such a function should not be confused with a
{\em digital contraction}~\cite{Bx94}, a homotopy between
an identity map and a constant function.

\begin{definition}
\label{KannanDef}
If         \[ d(f(x),f(y)) \le \alpha [d(x, f(x)) + d(y, f(y))] \]
          for all $x,y \in X$, where $0 < \alpha < 1/2$, we say
          $f$ is a {\em Kannan contraction map}.
\end{definition}

\begin{definition}
\label{ChatterjeaDef}
    If
          \[ d(f(x),f(y)) \le \alpha [d(x, f(y)) + d(y, f(x))] \]
          for all $x, y \in X$, where $0 < \alpha < 1/2$, we say
          $f$ is a {\em Chatterjea contraction map}.
    \end{definition}

\begin{definition}
\label{ReichDef}
If
\[ d(f(x),f(y)) \le ad(x, f(x)) + bd(y, f(y)) + cd(x, y)
\]
          for all $x,y \in X$ and all nonnegative $a,b,c$ such that
          $a+b+c < 1$, then $f$ is a {\em Reich contraction map}.
\end{definition}

\begin{prop}
\label{ReichIsContraction}
A Reich contraction map is a digital contraction map and is a
Kannan contraction map.
\end{prop}

\begin{proof}
\label{ReichParticularizes}
Let $f$ be a Reich contraction map.
That $f$ is a digital contraction map follows from the observation
of~\cite{Reich} that in Definition~\ref{ReichDef}, we can take
$a=b=0$ and obtain the conclusion from
Definition~\ref{contractionDef}. That $f$ is a Kannan contraction
map follows from the observation that in Definition~\ref{ReichDef},
we can take $a=b \in (0,1/2)$ and $c=0$
to obtain the conclusion from Definition~\ref{KannanDef}.
\end{proof}

\begin{definition}
\label{Zamf}
\cite{Mishra-etal}
Let $(X,d,\kappa)$ be a digital metric space and let
$f: X \to X$ be a function. If there exists
$\alpha \in (0,1)$ such that for all $x,y \in X$ we have
\[ d(f(x),f(y)) \le \alpha \, \max\{d(x,y),
   \frac{d(x,f(x)) + d(y,f(y))}{2},
   \frac{d(x,f(y)) + d(y,f(x))}{2}\}
\]
then $f$ is called a {\em Zamfirescu digital contraction}.
\end{definition}

\begin{definition}
\label{Rhoades}
\cite{Mishra-etal}
Let $(X,d,\kappa)$ be a digital metric space and let
$f: X \to X$ be a function. If there exists
$\alpha \in (0,1)$ such that for all $x,y \in X$ we have
\[ d(f(x),f(y)) \le \alpha \, \max\{d(x,y),
   \frac{d(x,f(x)) + d(y,f(y))}{2},
   d(x,f(x)), d(y,f(y))\}
\]
then $f$ is called a {\em Rhoades digital contraction}.
\end{definition}

\begin{prop}
\label{ZamfAndRhoadesContractions}
Let $(X,d,\kappa)$ be a digital metric space and let
$f: X \to X$ be a function. If $f$ is a Zamfirescu digital
contraction or a Rhoades digital contraction, then $f$
is a digital contraction map.
\end{prop}

\begin{proof}
From Definitions~\ref{Zamf} and \ref{Rhoades},
$d(f(x), f(y)) \le \alpha d(x,y)$ for all $x,y \in X$. The
assertion follows from Definition~\ref{contractionDef}.
\end{proof}

The following is a minor generalization of a definition
in~\cite{Hossain-etal}. Therefore, results we derive in this
paper for digitally $(\alpha, \kappa)$-uniformly locally
contractive functions apply to the version
in~\cite{Hossain-etal}.

\begin{definition}
\label{locallyContractive}
Suppose $0 \le \alpha < 1$. Let $(X,d,\kappa)$ be a digital
metric space. Let $f: X \to X$ be a function such that
$d(x,y) \le 1$ implies $d(f(x),f(y)) \le \alpha d(x,y)$.
Then $f$ is called
{\em digitally $(\alpha, \kappa)$-uniformly locally
contractive}.
\end{definition}

\begin{prop}
A digital contraction map with multiplier $\alpha$ is a
digitally $(\alpha, \kappa)$-uniformly locally
contractive map.
\end{prop}

\begin{proof}
This is obvious from Definition~\ref{contractionDef} and
Definition~\ref{locallyContractive}.
\end{proof}

Below, we define a set of functions $\Psi$ that will
be used in the following.

\begin{definition}
{\rm \cite{JR18}}
\label{psiFamily}
Let $\Psi$ be a set of functions
$\psi: [0,\infty) \to [0,\infty)$ such that for each
$\psi \in \Psi$ we have
\begin{itemize}
    \item $\psi$ is nondecreasing, and
    \item there exists $k_0 \in \N$, $a \in (0,1)$, and a
          convergent series $\sum_{k=1}^{\infty} v_k$ of
          non-negative terms such that $k \ge k_0$ implies
          $\psi^{k+1}(t) \le a \psi^k(t) + v_k$ for all
          $t \in [0,\infty)$, where $\psi^k$ represents the
          $k$-fold composition of $\psi$.
\end{itemize}
\end{definition}

The following will be used later in the paper.

\begin{exl}
\label{0inPsi}
The constant function with value 0 is a member of~$\Psi$.
\end{exl}

\begin{definition}
{\rm \cite{SametEtAl}}
\label{alphaAdmissible}
Let $T: X \to X$ and $\alpha: X \times X \to [0, \infty)$.
We say $T$ is {\em $\alpha$-admissible} if
$\alpha(x,y) \ge 1$ implies $\alpha(T(x),T(y)) \ge 1$.
\end{definition}

\begin{definition}
{\rm \cite{SametEtAl}}
\label{alpha-psi-contractive}
Let $(X,d)$ be a metric space, $T: X \to X$,
$\alpha: X \to X$, and $\psi \in \Psi$. We say $T$ is an
{\em $\alpha-\psi$-contractive mapping} if
$\alpha(x,y) d(T(x),T(y)) \le \psi(d(x,y))$
for all $x,y \in X$.
\end{definition}

\begin{remark}
{\rm \cite{JR18}}
\label{contractionIsAlphaPsiContractive}
A digital contraction map $f: (X,d,\kappa) \to (X,d,\kappa)$
is an $\alpha-\psi$-contractive mapping for $\alpha(x,y)=1$
and $\psi(t) = \lambda t$ for $\lambda \in (0,1)$.
\end{remark}

\begin{definition}
\label{expansive3}
{\rm \cite{JR18}}
Let $(X,d,\kappa)$ be a digital metric space, $T: X \to X$,
$\beta: X \times X \to [0,\infty)$, and
$\psi, \phi \in \Psi$ such that
\[ \psi(d(T(x), T(y))) \ge \beta(x,y) \psi(d(x,y)) +
                           \phi(d(x,y))
\]
for all $x,y \in X$. Then $T$ is a
{\em $\beta - \psi - \phi$-expansive mapping}.
\end{definition}

Depending on the choice of functions $\beta, \phi$ in
Definition~\ref{expansive3}, the definition may not be
very discriminating, as we see in the following.

\begin{prop}
\label{expansiveNondistinguishing}
Every function $T: X \to X$ is a
$\beta - \psi - \phi$-expansive mapping
if we take $\beta$ and $\phi$ to be
constant functions with value 0.
\end{prop}

\begin{proof}
The assertion follows from Example~\ref{0inPsi} and
Definition~\ref{expansive3}.
\end{proof}

\begin{definition}
{\rm \cite{Dol-Nal}}
\label{weaklyUnif}
Let $(X,d,\kappa)$ be a digital metric space.
Let $T: X \to X$. Then $T$ is a
{\em weakly uniformly strict digital contraction} if given
$\varepsilon > 0$ there exists $\delta > 0$ such that
$\varepsilon < d(x,y) < \varepsilon + \delta$ implies
$d(T(x),T(y)) < \varepsilon$ for all $x,y \in X$.
\end{definition}

\begin{definition}
{\rm \cite{JR18}}
\label{expansive}
Let $(X,d,\kappa)$ be a complete digital metric space.
Let $T: X \to X$. If $T$ satisfies the condition
$d(T(x),T(y)) \ge k d(x,y)$ for all $x,y \in X$ and
some $k >1$, then $T$ is a {\em digital expansive mapping}.
\end{definition}

\begin{exl}
\label{expansiveExl}
The function $T: \N \to \N$ defined by $T(n)=2n$ is a
digital expansive mapping, using the usual Euclidean metric.
This map is not $c_1$-continuous~\cite{Rosenfeld87}.
\end{exl}

The literature contains the following theorems concerning
fixed points for such functions.

The following is a digital
version of the Banach contraction principle~\cite{Banach}.

\begin{thm}
\label{Banach}
{\rm \cite{EgeKaraca}}
Let $(X,d,\kappa)$ be a complete digital metric space,
where $d$ is the Euclidean metric in $\Z^n$. Let $f: X \to X$ be a digital
contraction map. Then $f$ has a unique fixed point.
\end{thm}

The following is a digital version
of the Kannan fixed point theorem~\cite{Kannan}.

\begin{thm}
{\rm \cite{Ege-etal}}
\label{Kannan}
Let $f: X \to X$ be a Kannan contraction map on a digital metric space
$(X,d,\kappa)$. Then $f$ has a unique fixed point in $X$.
\end{thm}

The following is a digital version of the
Chatterjea fixed point theorem~\cite{Chatterjea}.

\begin{thm}
{\rm \cite{Ege-etal}}
\label{Chatterjea}
If $f: X \to X$ is a Chatterjea contraction map on a digital metric
space $(X,d,\kappa)$, then $f$ has a unique fixed point.
\end{thm}

The following gives digital
versions of Zamfirescu~\cite{Zamf72} and
Rhoades~\cite{Rhoades} fixed point theorems.

\begin{thm}
{\rm \cite{Mishra-etal}}
\label{ZamfAndRhoadsFixed}
Let $d$ be the Euclidean metric on $\Z^n$ and let
$(X,d,\kappa)$ be a digital metric space.
If $f: X \to X$ is a Zamfirescu digital contraction or a
Rhoades contraction map, then $f$ has a unique fixed point.
\end{thm}

The following is a digital version of the Reich fixed point
theorem~\cite{Reich}. We give a simpler proof of the
digital version than appeared in~{\rm \cite{Ege-etal}}.

\begin{thm}
\label{Reich}
Let $f: (X,d,\kappa) \to (X,d,\kappa)$ be a Reich
contraction map on a digital metric space. Then $f$ has a
unique fixed point in $X$.
\end{thm}

\begin{proof}
This follows immediately from
Proposition~\ref{ReichIsContraction} and
Theorem~\ref{Banach}.
\end{proof}

The following is a version of the Edelstein fixed point
theorem~\cite{Edelstein} for digital images.

\begin{thm}
\label{HossainFixed}
{\rm \cite{Hossain-etal}}
A digitally $(\alpha, \kappa)$-uniformly locally
contractive function on a connected complete digital
metric space has a unique fixed point.
\end{thm}

\section{Digital homotopy fixed point theory}
The paper~\cite{EgeKaraca-a} defines a
{\em digital fixed point property} as follows.
The digital image $(X,\kappa)$ has the digital fixed
point property with respect to the digital interval
$[0,m]_{\Z}$ if for all $(\kappa_*,\kappa)$-continuous
functions $f: (X \times [0,m]_{\Z},\kappa_*) \to (X,\kappa)$,
where $\kappa_*$ is the normal product adjacency for
$(X,\kappa) \times ([0,m]_{\Z}, 2)$,
there is a $\kappa$-path
$p: [0,m]_{\Z} \to X$ of fixed points, i.e., $p(t)$ is a
fixed point of the induced function
$f_t: X \to X$ defined by $f_t(x)=f(x,t)$.

Also,~\cite{EgeKaraca-a} defines a
{\em digital homotopy fixed point property} and states
that this is equivalent to the following: A digital image
$(X,\kappa)$ has the digital homotopy fixed point
property if for each digital homotopy
$f: X \times [0,m]_{\Z} \to X$ there is a $\kappa$-path
$p: [0,m]_{\Z} \to X$ such that for all $t \in [0,m]_{\Z}$,
$p(t)$ is a fixed point of the induced function
$f_t$.

These imply triviality, as follows.

\begin{thm}
A digital image $(X,\kappa)$ has the digital
fixed point property and the digital homotopy
fixed point property if and only if $X$ is a singleton.
\end{thm}

\begin{proof}
Clearly a singleton has the digital homotopy fixed
point property and the digital homotopy
fixed point property. Conversely, if $X$ is not a singleton,
then by Theorem~\ref{singleton}
there is a continuous function
$f: X \to X$ that does not have a fixed point. Let
$F: X \times [0,m]_{\Z} \to X$ be defined by $F(x,t)=f(x)$.
Then $F$ is both
$(\kappa_*,\kappa)$-continuous
(where $\kappa_* = \kappa_*(\kappa,2)$ is the normal product
adjacency) and a homotopy, and
fails to have a fixed point for any of the
induced functions $f_t=f$. Thus, $(X,\kappa)$ does not have
the digital fixed point property or the digital homotopy
fixed point property.
\end{proof}

\section{Results for various contraction and expansion maps}
\subsection{Digital contraction maps}
In both of the papers \cite{EgeKaraca,Hossain-etal},
arguments are given for the incorrect assertion
that every digital contraction map is
digitally continuous. Both papers present an error of
confusing (topological) continuity of a map between
metric spaces with (digital) continuity of a map between
digital images. We present a
counterexample to this assertion;
our example is also used to show that Kannan,
Chatterjea, Zamfirescu, and Rhoades contraction maps
need not be digitally continuous. We use the
Manhattan metric for its ease
of computation, but the Euclidean or other $\ell_p$ metrics could be used to obtain similar conclusions.

\begin{exl}
\label{counter}
Let
\[ X = \{p_1 = (0,0,0,0,0), ~~ p_2 = (2,0,0,0,0), ~~ p_3 = (1,1,1,1,1)\} \subset \Z^5. \]
Let $f: X \to X$ be defined by $f(p_1)=f(p_2) = p_1$, $f(p_3)=p_2$.
Then $f$ is not $(c_5,c_5)$-continuous. However,
with respect to the Manhattan metric $d$,
$f$ is
\begin{itemize}
    \item a digital contraction map,
    \item a Kannan contraction map,
    \item a Chatterjea contraction map,
    \item a Zamfirescu contraction map,
    \item a Rhoades contraction map,
    \item a $(0.45, c_5)$-uniformly local contraction,
    \item an $\alpha-\psi$-contractive mapping for
          $\alpha(x,y)=1$ and $\psi(t) = \lambda t$ for
          $\lambda \in (0,1)$,
    \item a $\beta - \psi - \phi$-expansive mapping, where
          $\psi$ and $\phi$ are constant functions with the
          value 0,
    \item a weakly uniformly strict digital contraction.
\end{itemize}
\end{exl}

\begin{proof}
Note $f$ is not $(c_5,c_5)$-continuous, since
$p_1 \adj_{c_5} p_3 \adj_{c_5} p_2$, but $f(X) = \{p_1,p_2\}$ is not
$c_5$-connected.

Observe that
\[ d(p_1,p_2)=2, ~~~ d(f(p_1),f(p_2)) = 0,
\]
\[ d(p_1,p_3) = 5, ~~~ d(f(p_1),f(p_3)) = 2,\]
\[ d(p_2,p_3)= 5, ~~~ d(f(p_2),f(p_3)) = 2.\]
Therefore, we have, for all $x,y \in X$ such that $x \ne y$,
$d(f(x),f(y)) \le 2/5 \, d(x,y) < 0.45 d(x,y)$. Therefore, $f$ is a digital contraction map, a Zamfirescu contraction
map, a Rhoades contraction map, and a $(0.45, c_2)$-uniformly
local contraction.

Since
\[ d(f(x),f(y)) \le 2/5 [d(x, f(x)) + d(y, f(y))] <
   0.45 [d(x, f(x)) + d(y, f(y))]
\]
for all $x,y \in X$ such that $x \ne y$, $f$ is a Kannan contraction map.

Note
\[ d(f(p_1),f(p_2)) = 0 < 0.45 [d(p_1,f(p_2)) + d(p_2,f(p_1))],
\]
\[ d(f(p_1,f(p_3)) = 2 < 0.45 (2 + 5) = 0.45 [d(p_1,f(p_3)) + d(p_3,f(p_1)),]
\]
\[ d(f(p_2),f(p_3)) = 2 < 0.45 (0 + 5) = 0.45 (d(p_2,f(p_3)) + d(p_3,f(p_2)).
\]
Therefore, $f$ is a Chatterjea contraction map.

That $f$ is an $\alpha-\psi$-contractive mapping for
$\alpha(x,y)=1$ and $\psi(t) = \lambda t$ for
$\lambda \in (0,1)$ follows from
Remark~\ref{contractionIsAlphaPsiContractive}.

Since Example~\ref{0inPsi} notes that the constant function
with value 0 is in~$\Psi$, it follows from
Definition~\ref{expansive3} that $f$ is a
$\beta - \psi - \phi$-expansive mapping.

It follows easily from Definition~\ref{weaklyUnif} that
$f$ is a weakly uniformly strict digital contraction.
\end{proof}

The following generalizes Theorem~4.7(1) of~\cite{Han16}.
We give a proof, essentially that of~\cite{Han16}, so we
can refer to it below.

\begin{thm}
\label{c1}
Let $(X,d,c_1)$ be a digital metric space that is $c_1$-connected,
where $d$ is any $\ell_p$ metric in $\Z^n$. Let $f: X \to X$ be a digital
contraction map. Then $f$ is a constant function.
\end{thm}

\begin{proof}
Let $\alpha \in (0,1)$ satisfy $d(f(x), f(y)) \le \alpha d(x,y)$ for all
$x,y \in X$. If $x \adj_{c_1} y$ in $X$, then $d(x,y)=1$, so
$d(f(x), f(y)) \le \alpha$, which implies $f(x)=f(y)$, since every distinct pair
of points in $\Z^n$ has distance of at least 1.

Given $x_0 \in X$,
for any $x \in X$ there is a path $P=\{x_0, x_1, \ldots, x_m = x\} \subset X$ from
$x_0$ to $x$ such that $x_i \adj_{c_1} x_{i+1}$, $0 \le i <m$. It follows from
the above that $f$ is the constant function with value $f(x_0)$.
\end{proof}

Theorem~4.7(2) of~\cite{Han16} gives examples of $c_2$-connected images
with digital contraction maps that are continuous and
not constant. However, modification
of Theorem~\ref{c1} yields the following.

\begin{thm}
\label{cu}
Let $(X,d,\kappa)$ be a digital metric space that is $\kappa$-connected, where, for some $M_1 \ge M_2 > 0$
we have that
$x \ne y$ implies $d(x,y) \ge M_2$ and
$x \adj_{\kappa}y$ implies $d(x,y) \le M_1$
Let $f: X \to X$ be a digital contraction map with
multiplier $\alpha$ such that $\alpha < M_2 / M_1$.
Then $f$ is a constant function.
\end{thm}

\begin{proof}
Let $x,y \in X$ such that $x \adj_{\kappa} y$. Then
$d(x,y) \le M_1$ and
\[d(f(x),f(y)) < \alpha d(x,y)< (M_2 / M_1) M_1 = M_2.\]
By choice of $M_2$, $f(x)=f(y)$.
It follows as in the proof of
Theorem~\ref{c1} that $f$ is constant.
\end{proof}

\begin{remark}
{\rm
Notice that Theorem~\ref{cu} applies to all connected
digital images $(X,d,c_u)$, where $X \subset \Z^n$,
$1 \le u \le n$, and $d$ is any $\ell_p$ metric.
}
\end{remark}

\begin{remark}
{\rm
It follows from Theorem~\ref{cu} and
Proposition~\ref{ZamfAndRhoadesContractions} that
if $(X,d,c_u)$ is a digital metric space that is $c_u$-connected,
where $X \subset \Z^n$, $1 \le u \le n$,
$d$ is any $\ell_p$ metric in $\Z^n$, and $f: X \to X$ is a
Zamfirescu or Rhoades
contraction map such that for all $x, y \in X$,
$d(f(x),f(y)) \le \alpha d(x,y)$, where $0 < \alpha < 1/\sqrt[p]{u}$, then $f$ is a constant function.
}
\end{remark}

\subsection{Kannan and Chatterjea contractions}
Example~\ref{counter} shows that neither a Kannan contraction map
nor a Chatterjea contraction map must be constant.
However, we have the following.

\begin{thm}
\label{KanChatConst}
Let $(X,d,\kappa)$ be a digital metric space
of finite diameter, where $d$ is any $\ell_p$ metric.
Let $f: X \to X$ be a function. If $f$ is a Kannan contraction
map or a Chatterjea contraction map with $\alpha$ as in
Definition~\ref{KannanDef} or Definition~\ref{ChatterjeaDef},
respectively, satisfying
$0 < \alpha < \frac{1}{2 \, diam X}$, then $f$ is a constant
function.
\end{thm}

\begin{proof}
We have
$d(f(x), f(y)) < 1$ for all $x,y \in X$, by
Definition~\ref{KannanDef} in the case of a Kannan contraction map,
and by Definition~\ref{ChatterjeaDef} in the case of a Chatterjea
contraction map. Since $d$ is an $\ell_p$
metric, it follows that $f(x)=f(y)$ for all $x,y \in X$.
\end{proof}

\subsection{Reich contractions}
\begin{thm}
\label{ReichConst}
Let $(X,d,\kappa)$ be a digital metric space of positive, finite diameter,
where $d$ is any $\ell_p$ metric.
Let $f: X \to X$ be a function. If $f$ is a Reich contraction
map with $a,b,c$ as in Definition~\ref{ReichDef} satisfying
$a,b,c \in (0, \frac{1}{3 \, diam X})$, then $f$ is a constant
function.
\end{thm}

\begin{proof}
We have
$d(f(x), f(y)) < 1$ for all $x,y \in X$. Since $d$ is an $\ell_p$
metric, it follows that $f(x)=f(y)$ for all $x,y \in X$.
\end{proof}

Also, it follows from Proposition~\ref{ReichIsContraction}
that Theorem~\ref{c1} and
Theorem~\ref{cu} apply to a Reich contraction map.

\subsection{$(\alpha, \kappa)$-uniformly locally
contractive functions}
Theorem~\ref{HossainFixed} turns out to be a trivial
result for connected digital metric spaces that use an
$\ell_p$ metric, as we see below.

\begin{thm}
Let $(X,d,\kappa)$ be a digital metric space,
where $d$ is any $\ell_p$ metric. Let
$f: X \to X$ be an $(\alpha, \kappa)$-uniformly
locally contractive function. Then $f$ is a constant function.
\end{thm}

\begin{proof}
Let $x_0, x \in X$. Since $X$ is connected, there
is a $\kappa$-path in $X$, $\{x_i\}_{i=0}^m$, from $x_0$ to
$x$ such that $x_m=x$ and $x_i \adj_{c_1} x_{i+1}$
for $0 \le i < m$. If $d(x_i, x_{i+1}) \le 1$, then
\[ d(f(x_i),f(x_{i+1})) \le \alpha d(x_i, x_{i+1}) < 1.
\]
Since $d$ is an $\ell_p$ metric, $f(x_i) = f(x_{i+1})$.
It follows that $f$ is a constant function.
\end{proof}

\subsection{Digital expansive mappings}
We saw in Example~\ref{expansiveExl} that a digital
expansive mapping need not be digitally continuous.

Theorem~3.2 and Corollary~3.3,
of~\cite{JR17} hypothesize
a digital expansive mapping $T: X \to X$ that is onto.
But in ``real world" image processing, a digital image
is finite, and therefore cannot support such a map,
as shown by the following Theorems~\ref{notOnto1}
and~\ref{notOnto2}.

\begin{thm}
\label{notOnto1}
Let $(X,d,\kappa)$ be a digital metric space of more
than one point. If $X$
has a bounded diameter, then there is no self-map
$T: X \to X$ that is onto and a digital expansive mapping.
\end{thm}

\begin{proof}
Suppose there is a digital expansive mapping
$T: X \to X$. Let $x_0,y_0 \in X$ be such that
$d(x_0,y_0) = diam X > 0$. Then for some $k >1$,
\begin{equation}
\label{expansiveEq}
     d(T(x_0),T(y_0)) \ge k d(x_0,y_0) = k\,  diam X > diam X.
\end{equation}
Since statement~(\ref{expansiveEq}) is
contradictory, the assertion follows.
\end{proof}

\begin{thm}
\label{notOnto2}
Let $(X,d,\kappa)$ be a digital metric space of more
than one point. If there exist $x_0,y_0 \in X$
such that
\begin{equation}
\label{minEq}
d(x_0,y_0) = \min\{d(x,y) \, | \, x,y \in X, d(x,y)>0\},
\end{equation}
then there is no self-map
$T: X \to X$ that is onto and a digital expansive mapping.
\end{thm}

\begin{proof}
Suppose there exists a map $T: X \to X$ that is onto
and a digital expansive mapping. Let $x_0,y_0 \in X$ be
as in equation~(\ref{minEq}).
Since $T$ is onto, there exist $x',y' \in X$ such that
$T(x')=x_0$, $T(y')=y_0$. Let $k$ be the expansive constant
of $T$. Then
\[ d(x_0,y_0) = d(T(x'),T(y')) \ge k d(x', y').
\]
Hence $d(x',y') \le d(x_0,y_0) / k < d(x_0,y_0)$,
which contradicts our choice of $x_0,y_0$.
The assertion follows.
\end{proof}

Note Theorem~\ref{notOnto2} is applicable when $X$ is finite
or when $d$ is any $\ell_p$ metric.

\begin{remark}
{\rm
Example~3.8 of~\cite{JR17} claims that the self-map
$T: X \to X$ of some subset of $\Z$ given by
$T(n)=2n-1$ is onto. It is easy to see that this
claim is only true for $X = \{1\}$.
}
\end{remark}

\subsection{$\beta - \psi - \phi$-expansive
mappings}
An analog of Theorem~2.1 of~\cite{SametEtAl} is asserted as Theorem~3.2 of~\cite{JR18}:

    {\em Let $(X,d,\kappa)$ be a complete digital metric space
    and $T: X \to X$. If there exist functions
    $\beta: X \times X \to [0,\infty)$ and
    $\psi,\phi \in \Psi$ such that
    \begin{itemize}
        \item $T^{-1}$ is $\beta$-admissible;
        \item there exists $x_0 \in X$ such that
              $\beta(x_0, T^{-1}(x_0)) \ge 1$; and
        \item $T$ is digitally continuous,
    \end{itemize}
    then $T$ has a fixed point.}

However, this assertion is false, as we see in the
following.

\begin{exl}
\label{noFix}
Let $X = [-1,1]_{\Z}^2 \setminus \{(0,0)\} \subset \Z^2$.
Let $\beta = d$ be the Manhattan metric on $\Z^2$.
Let $\phi$ and $\psi$ be constant functions with value 0.
Let $f: X \to X$ be the map defined by $f(x,y)=(-x,-y)$.
Then $f$ is $\beta$-admissible; for every $p \in X$ we
have $\beta(p, f^{-1}(p)) = d(p, -p) > 1$; and $f$ is
both $c_1$-continuous and $c_2$-continuous, but $f$ has no
fixed point.
\end{exl}

\begin{proof}
It was observed in Example~\ref{0inPsi} that the constant
function with value 0 is a member of $\Psi$. It is easy to
see that the assertion follows.
\end{proof}

The following is given as Theorem~3.3 (and, with
another hypothesis, as Theorem~3.7) of~\cite{JR18}.

\begin{thm}
\label{vacuous}
Let $(X,d,\kappa)$ be a complete digital metric space and
let $T: X \to X$ be $\beta - \psi - \phi$-expansive
mapping such that for some sequence
$\{x_n\}_{n=1}^{\infty} \in X$ we have
$\beta(x_n,x_{n+1}) \ge 1$ for all $n$ and
$x_n \rightarrow y \in X$ as $n \rightarrow \infty$, then
there is a subsequence $\{x_{n_k}\}$ of
$\{x_n\}_{n=1}^{\infty}$ such that
$\beta(x_{n_k},y) \ge 1$ for all $k$.
Then $T$ has a fixed point.
\end{thm}

However, we have the following.

\begin{exl}
If $X$ is finite or $d$ is an $\ell_p$
metric, and $\beta = d$, then
Theorem~\ref{vacuous} is vacuously true, as no such
sequence $\{x_n\}_{n=1}^{\infty} \in X$ exists.
\end{exl}

\begin{proof}
By Corollary~\ref{Han-Cauchy-cor}, $x_n \rightarrow y$
implies that for some $k_0$, $k>k_0$ implies
$x_n=y$ and therefore
$\beta(x_n,y)=d(x_n,y)=0$. Thus, no sequence
$\{x_n\}_{n=1}^{\infty}$ satisfies the hypotheses of
Theorem~\ref{vacuous}.
\end{proof}

\begin{remark}
{\rm
The following assertion is stated as Theorem~3.6
of~\cite{JR18}.}

Let $(X,d,\kappa)$ be a complete digital metric
space. Let $T: X \to X$ be a $\beta - \psi - \phi$-expansive
mapping such that
\[ \psi(d(T(x),T(y))) \ge \beta(x,y) \psi(M(x,y)) +
                          \phi(M(x,y))
\]
for all $x,y \in X$, where
\[ M(x,y) = \max \{d(x,y), d(x,T(x)), d(y,T(y)),
                   \frac{d(x,T(y)) + d(y, T(x))}{2}\}.
\]
Then $T$ has a fixed point.

{\rm But this assertion is false.
}
\end{remark}

\begin{proof}
It is easy to see that the choices of $X,d,\kappa,T,\beta,\psi,\phi$ of
Example~\ref{noFix} provide a counterexample to the assertion.
\end{proof}

\subsection{Remarks on~\cite{Dol-Nal}}
The paper~\cite{Dol-Nal} identifies one of the authors
of the current paper, L. Boxer, as a reviewer. In fact, errors and other
shortcomings mentioned in Boxer's review
remain in the published version of~\cite{Dol-Nal}.

The assertion stated as Theorem~3.1
of~\cite{Dol-Nal} is the following.
\begin{quote}
    {\em Let $(X,d,\kappa)$ be a complete
         metric space such that $T: X \to X$
         satisfies $d(T(x),T(y)) \le \psi(d(x,y))$ for all $x,y \in X$,
         where $\psi: [0,\infty) \to [0,\infty)$ is monotone
         nondecreasing and
         $\psi^n(t) \rightarrow 0$ as
         $n \rightarrow \infty$. Then $T$
         has a unique fixed point.
    }
\end{quote}

The argument offered in proof of this assertion confuses topological
continuity (the ``$\varepsilon - \delta$ definition")
and digital continuity (preservation of connectedness) in order to conclude that
$T$ is continuous. However, in
Example~\ref{counter}, using $\psi(t)=t/2$,
we have a function that satisfies the
hypotheses above and is not digitally
continuous.

A similar flaw appears in the argument given
in proof of the assertion states as
Theorem~3.3 of~\cite{Dol-Nal}, where again
Example~\ref{counter} provides a counterexample to the claim that a
weakly uniformly strict digital contraction mapping is digitally continuous.

Therefore, we must regard the assertions stated as Theorems~3.1 and~3.3
of~\cite{Dol-Nal}
as unproven. Since these and assertions dependent
on these make up all of the new assertions of the paper,
we conclude that nothing new is correctly
established in~\cite{Dol-Nal}.

\section{Common fixed points of intimate maps}
The paper~\cite{Jain} obtains a result
for common fixed points of intimate maps.
We show in this section that the
characterization of intimate maps given
in~\cite{Jain} can be simplified, and that
the primary result of~\cite{Jain} is
rather limited.

\begin{definition}
{\rm \cite{Jain}}
\label{intimateDef}
Let $(X,d,\kappa)$ be a digital metric
space. Let $f,g: X \to X$. Let $\alpha$
be either the $lim \, inf$ or the
$lim \, sup$ operation. If for
every $\{x_n\}_{n=1}^{\infty} \subset X$
such that
\begin{equation}
\label{intimateLim}
\lim_{n \to \infty} f(x_n) =
   \lim_{n \to \infty} g(x_n) = t
\end{equation}
for some $t \in X$ we have,
for $n$ sufficiently large,
\begin{equation}
\label{intimateEq}
     \alpha \, d(g(f(x_n)), g(x_n)) \le
   \alpha \, d(f(f(x_n)), f(x_n))
\end{equation}
then we say $f$ is {\em $g$-intimate}.
\end{definition}

\begin{prop}
Let $(X,d,\kappa)$ be a digital metric
space, where $d$ is any $\ell_p$ metric.
Let $f,g: X \to X$. Then $f$ is
$g$-intimate if and only if for every
sequence $\{x_n\}_{n=1}^{\infty} \subset X$
satisfying statement~(\ref{intimateLim})
we have, for $n$ sufficiently large,
\[ d(g(t), t) \le d(f(t),t).
\]
\end{prop}

\begin{proof}
From Theorem~\ref{Han-Cauchy}, a
sequence $\{x_n\}_{n=1}^{\infty} \subset X$
satisfying statement~(\ref{intimateLim}) has,
for $n$ sufficiently large,
$f(x_n)=g(x_n)=t$. The assertion follows
easily.
\end{proof}

\begin{thm}
{\rm ~\cite{Jain}}
\label{jainThm}
If $(X,d,\kappa)$ is a digital
    metric space and
    $A,B,S,T: X \to X$ are such that

    a) $S(X) \subset B(X)$ and
              $T(X) \subset A(X)$;

    b) for some $\alpha \in (0,1)$ and
              all $x,y \in X$,
    \begin{equation}
    \label{STFineq}
    d(S(x),T(y)) \le \alpha \, F(x,y)
    \end{equation}
    where
\[  F(x,y) =
   \max \{d(A(x),B(y)), d(A(x),S(x)),
   d(B(y),T(y)),\]
\[   ~~~d(S(x),B(y)), d(A(x),T(y))\};
\]

c) $A(X)$ is complete;

d) $S$ is $A$-intimate and
          $T$ is $B$-intimate,

\noindent  then $A$, $B$, $S$, and $T$ have a
    unique common fixed point.
\end{thm}

But Theorem~\ref{jainThm} is limited, as shown
by the following.

\begin{prop}
Suppose we assume the hypotheses of
Theorem~\ref{jainThm}, with $d$ being
an $\ell_p$ metric. Suppose
\begin{equation}
\label{alphaBound}
\alpha < \min \{1 / F(x,y) \, | \, x,y \in X \}
\end{equation}
or
\begin{equation}
\label{diamBound}
diam(S(X) \cup T(X)) = diam \, X < \infty.
\end{equation}
Then $S$ and $T$ are constant functions that
have the same value, and, in
case~(\ref{diamBound}), $X$ is a singleton.
\end{prop}

\begin{proof}
Inequality~(\ref{alphaBound}) and
hypothesis b) of Theorem~\ref{jainThm}
imply that $d(S(x),T(y)) < 1$, hence
$S(x)=T(y)$ for all $x, y \in X$.
Since $d$ is an $\ell_p$
metric, we have, for some $x_0, y_0 \in X$,
$S(x_0)=T(y)$ and $S(x)=T(y_0)$ for all
$x, y \in X$. Hence $S$ and $T$ are constant
functions with the same values.

Statement~(\ref{diamBound}) implies there
exist $x_0,y_0 \in X$ such that
$d(S(x_0),T(y_0)) = diam \, X$. Since
$F(x_0,y_0) \le diam \, X$,
inequality~(\ref{STFineq}) becomes
$diam \, X \le \alpha \, diam \, X$, which
implies $diam \, X = 0$. Thus, $X$ is a
singleton, so $S$ and $T$ are constant
functions with the same values.
\end{proof}

\section{Homotopy invariant fixed point theory}
It is natural in topology to consider the behavior of the fixed point set when a continuous function is changed by homotopy. In classical topology for nice spaces (for example the geometric realization of any finite simplicial complex), it is trivial to change a function by homotopy to increase the number of fixed points. The more interesting question is whether or not the number of fixed points can be decreased by homotopy.

The following Proposition~\ref{noFixed} was the key to
the proof of Theorem~\ref{singleton}.

\begin{prop}
{\rm \cite{Bx-etal}}
\label{noFixed}
Let $(X,\kappa)$ be a connected digital image of more than
one point. Let $x_0 \adj_{\kappa} x_1$ in $X$. Then the function $g: X \to X$ defined by
\[g(x) = \left \{ \begin{array}{cc}
    x_0 & \mbox{if } x \ne x_0; \\
     x_1 & \mbox{if } x = x_0,
\end{array} \right .
\]
is continuous and has no fixed points.
\end{prop}

\begin{prop}
\label{htpNoFixed}
Let $(X,\kappa)$ be a connected digital image of more than
one point. Then any constant map $f: X \to X$ is homotopic
to a map without fixed points.
\end{prop}

\begin{proof}
Let $x_0, x_1 \in X$ with $x_0 \adj_{\kappa} x_1$. Let
$f: X \to X$ be the constant map with image $\{x_0\}$.
Let $H: X \times [0,1]_{\Z} \to X$ be defined by
$H(x,0)= x_0$; $H(x,1)=x_0$ for $x \ne x_0$; $H(x_0,1)=x_1$.
It is easy to see that $H$ is a homotopy from $f$ to a
function $g$ as in Proposition~\ref{noFixed} without fixed points.
\end{proof}

Let $\MF(f)$ be the minimal number of fixed points among all continuous functions homotopic to $f$. For example, if $X$ has only 1 point then clearly $\MF(f) = 1$. If $X$ has more than 1 point and $(X,\kappa)$ is contractible, then any continuous function $f:X\to X$ is homotopic to a constant
function, and it follows from Proposition~\ref{htpNoFixed} that $\MF(f) = 0$.

Several examples are given in \cite{hmps} of digital images $(X,\kappa)$ for which no function on $X$ is homotopic to the identity except for the identity itself. Such images
are called \emph{rigid}. For example, a wedge product of two loops, each having at least 5 points, is rigid. Clearly if $X$ is a rigid digital image having $n$ points and $\id$ denotes the identity function, then $\MF(\id) = |X|$.

In classical topology, $\MF(f)$ can often be computed by Nielsen fixed point theory; see \cite{jiang}. Each fixed point is assigned an integer-valued \emph{fixed point index}, which can be computed homologically.
When $x$ is an isolated fixed point of $f$, the fixed point index of $x$ is denoted $\ind(f,x)$.
This index is a sort of multiplicity count for the fixed point: when $\ind(f,x)= 0$ then the fixed point at $x$ can be removed by a homotopy.

The fixed point index is homotopy invariant in the following sense: if, during some homotopy $f\simeq g$, the fixed point $x$ of $f$ moves into a fixed point $y$ of $g$, then $\ind(f,x) = \ind(g,x)$. Furthermore, when the fixed point set of $f$ is finite, then the sum of all the fixed point indices equals the Lefschetz number $L(f)$. In this sense $\ind(f,x)$ provides a localized version of the Lefschetz number.

In Nielsen fixed point theory, the fixed points are grouped into \emph{Nielsen classes}, and the number of such classes having nonzero index sum is the \emph{Nielsen number} $N(f)$. This number is a homotopy invariant satisfying $N(f) \le \MF(f)$, and in many cases (for example when $X$ is a manifold of dimension different from 2), $N(f) = \MF(f)$.

The 2012 paper~\cite{eklefschetz} by Ege \& Karaca attempts to develop a Lefschetz fixed point theorem for digital images, but the main result is incorrect, and was retracted in the 2016 paper \cite{Bx-etal}. The same authors attempted to develop a Nielsen theory in the 2017 paper \cite{EgeKaraca17} based on their faulty Lefschetz theory. The theory developed in \cite{EgeKaraca17} is also incorrect.

The main problem in \cite{EgeKaraca17} is inherited from problems in \cite{eklefschetz}, and concerns the definition of the fixed point index. Definition 3.2 of \cite{EgeKaraca17} does not give a satisfactory definition of the function $F$, and the degree used is inadequate because the appropriate homology groups are not necessarily $\Z$. The authors claim to define an integer valued Nielsen number $N(f)$ which is a homotopy invariant (Theorem 3.6) and a lower bound for $\MF(f)$ (Theorem 3.7).
Their Example 3.4, claiming that if $f$ is a
constant then $N(f)=1$, yields a contradiction for connected images $X$ such that $|X|>1$, since our Proposition~\ref{htpNoFixed} implies that $MF(f)=0$ for such functions $f$.

The errors in this work are not merely mistakes but indicate fundamental flaws in the theory. Anything resembling the standard homological definitions of $L(f)$ and the fixed point index will require that the Lefschetz number and fixed point index of the constant map equal 1. This cannot be reconciled with the fact that, when $X$ has more than 1 point, the constant map can be changed by homotopy to have no fixed points.

The authors believe that any successful theory for computing $\MF(f)$ will involve techniques very different from classical Lefschetz and Nielsen theory. The setting of digital images also allows the study of the quantity $\XF(f)$, the maximum number of fixed points among all functions homotopic to $f$. In classical topological fixed point theory this number is typically infinite, but for a digital image with $n$ points, clearly $\XF(f) \le n$ for any $f$. In fact our definition of $\XF(f)$ implies that $f$ is homotopic to the identity if and only if $\XF(f)=n$. When $f$ is a constant, then $1 \le \XF(f)\le n$, and for many choices of the image $X$ we will have $\XF(f) < n$. We do not know if it is possible for $\XF(f)=0$ for any function on a connected digital image.

Although many of the concepts discussed in this paper turn out to be trivial or otherwise uninteresting, the questions of computing $\MF(f)$ and $\XF(f)$ seem to be difficult and interesting, and present opportunities for further work.
Variations that count approximate fixed points would also be interesting objects of study.

\section{Concluding remarks}
Although the study of fixed points, or approximate fixed
points, is important in digital topology as in other
branches of mathematics, it does not appear that the use
of metric spaces yields useful knowledge in this area. We
have seen that metric space functions introduced to
study fixed points in digital topology - digital contraction
maps, Kannan contraction maps, Chatterjea contraction maps,
Zamfirescu contraction maps, Rhoades contraction maps,
Reich contraction maps, uniformly locally
contractive functions, intimate functions
- often turn out to be either discontinuous
or constant - hence, arguably uninteresting -
when the image considered is finite or when
common metrics are used.

It appears to us that the most natural metric function
to use for a connected digital image $(X,\kappa)$ is
the path length metric~\cite{Han05}:
$d(x,y)$ is the length of a shortest $\kappa$-path from
$x$ to $y$. Since this metric reflects
$\kappa$, it seems far superior to
an $\ell_p$ metric on a digital image. However, even this
metric gives us little new information. Since it is
integer-valued, its Cauchy sequences are also eventually
constant.

We have also corrected errors and pointed out trivialities
in other papers concerned with fixed points or approximate
fixed points of continuous self-maps of digital images.


TITLE: Why do we take the axiom of induction for natural numbers (Peano arithmetic)?
QUESTION [16 upvotes]: More precisely, when we define the set of natural numbers $\mathbb{N}$ using the Peano axioms, we assume the following:

$0\in\mathbb{N}$
$\forall n\in\mathbb{N} (S(n)\in\mathbb{N})$
$\forall n\in\mathbb{N}(0\neq S(n))$
$\forall m,n (m\neq n\to S(m)\neq S(n))$
If $P(n)$ denotes the fact that $n$ has property $P$, then $\Big(P(0)\wedge \forall n\in\mathbb{N}\big(P(n)\to P(S(n))\big)\Big)\implies \forall n\in \mathbb{N} (P(n))$

I understand that using these axioms we can derive everything about the natural numbers, but I also think it's helpful to know why the axioms were chosen the way they are. So my question is why we choose to accept the axiom of induction ((5.) above), which in a way makes this more of a metamathematical question.
For example in Tao's Analysis I, it says that the axiom of induction keeps unwanted elements (such as half-integers) from entering the set.
Wikipedia says, "Axioms [1], [2], [3] and [4] imply that the set of natural numbers is infinite, because it contains at least the infinite subset $\{ 0, S(0), S(S(0)), \ldots \}$, each element of which differs from the rest. To show that every natural number is included in this set requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers."---But I find this tautological: $\mathbb{N}$ is defined as the set of natural numbers so "$n$ is a natural number" means "$n\in\mathbb{N}$", right? So isn't every natural number included in  $\mathbb{N}$ by definition?
Suppose we want to show $\mathbb{N}=\{0,1,2,3,\ldots\}$ using all five of the Peano axioms.
If we let $P(n)$ denote $n\in\{0,1,2,3,\ldots\}$, then $P(0)$ is true. Suppose $n$ is in $\{0,1,2,3,\ldots\}$. Then (informally) the dots indicate that $S(n)$ is in $\{0,1,2,3,\ldots\}$. So $\mathbb{N}\subseteq\{0,1,2,3,\ldots\}$, i.e., our defined set contains no "extra" elements (as in Tao's Analysis I).
Yet I still do not see how to show $\{0,1,2,3,\ldots\}\subseteq\mathbb{N}$ (in order to complete the "proof" that $\mathbb{N}=\{0,1,2,3,\ldots\}$) without just assuming it. (I think this is what the Wikipedia article was doing(?))
Thanks in advance for any help and I apologize if this kind of question is unsuitable for this site.

REPLY [0 votes]: Note: I'm still learning about the Peano Axioms, so if any part of my answer is inaccurate, please let me know.
First, consider why we even bother thinking about the natural numbers. Of course, it's because they obey unique properties; namely, the natural numbers are "natural" in the sense that they represent how we count real-life objects. Clearly, the set of natural numbers is no ordinary set, so we might say that $\mathbb{N}$ is a "set with structure," where the "structure" encapsulates all the unique properties that make the natural numbers special.
Think of the Peano Axioms as a means of formulating this "structure." Any set obeying this "structure" should either be the natural numbers, or a set isomorphic to the natural numbers. Therefore, it doesn't make much sense to try and prove that $\{0, 1, 2, 3, \dots\} \subseteq \mathbb{N}$ (as you do in your answer); instead, you should be trying to prove that the set $\mathbb{N} = \{0, 1, 2, 3, \dots \}$ paired with the canonical successor function obeys the Peano Axioms, and thus represents the natural numbers.
If you view the axioms in this way, your question can be reframed as, "Why are the first four Peano Axioms insufficient in defining the 'structure' of $\mathbb{N}$?" A simple counterexample suffices to answer this, which Tao's half-integers does beautifully. Tao defines the half-integers to be
$$\mathbb{N} := \{0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, \dots\}.$$
and its successor function $S$ to be
$$
S : \mathbb{N} \to \mathbb{N} \\
S(n) = n + 1
$$
where $+$ is the standard real addition operator. It isn't difficult to check that this obeys the first four Peano Axioms. But this set is not the natural numbers, nor is it isomorphic to them! Therefore, we add a fifth axiom, which (in plain English) states that, "All natural numbers  are some eventual successor of $0$."


TITLE: Continuity of support functionals
QUESTION [5 upvotes]: Let $X$ be a normed space(not necessarily Banach) and let $G$ be a $w^*-$compact subset of $X^*,$ the dual space of $X.$ Consider the function $f:X\to \Bbb R$ defined by
$$f(x)= \max_{g^*\in G}\{\langle g^*,x\rangle\}.$$ Is $f$ continuous? If not, what about lower semicontinuous? Could you present a example of a discontinuous $f$?
I can prove that if $X$ is Banach the function is Lipschitz(by using the Uniform Boundedness Theorem). However, in the general case I am still curious.

REPLY [3 votes]: Is $f(x)$ well-defined?
Yes, if $G$ is not empty. Let $x\in X$ be given.
The function $\psi_x:X^*\to\mathbb R$, $x^*\mapsto \langle x^*, x\rangle$
is continuous from the $w^*$-topology of $X^*$ to $\mathbb R$.
Therefore, the image of $G$ under $\psi_x$ is compact.
Because the maximum of each non-empty compact subset of $\mathbb R$ exists and is real-valued,
$f(x)$ is well-defined.
Is $f$ continuous?
I think I found a counterexample.
Let $X=\ell^2_0$,
which is the subspace of $\ell^2$ which consists of sequences with only finitely many nonzero entries.
Clearly, $X$ is not a Banach space and $\ell^2$ is its dual space.
We set
$$
G := \{\sqrt{n}e_n : n\in\mathbb N\} \cup \{0\}\subset X^*.
$$
Then it can be shown that $G$ is a $w*$-compact subset of $X^*$
(I can fill in the details if requested).
Let us show that $f$ is not continuous.
Let $x_n:=n^{-1/2}e_n$ be a sequence in $\ell^2_0$.
Clearly, $x_n\to 0$.
However, we have $f(x_n)=\langle \sqrt{n} e_n,n^{-1/2}e_n\rangle = 1$
and $f(0)$, but not $f(x_n)\to f(0)$.
Therefore $f$ is not continuous.
This seems to contradict the answer of @Red shoes.
Is $f$ lower semicontinuous?
Yes, $f$ should be lower semicontinuous.
This is because of the arguments in the answer of @Red shoes.
Under what conditions on $G$ is $f$ continuous?
It can be shown that $f$ is continuous if and only if $G$ is bounded
(and if and only if $f$ is continuous at $0$).
First, suppose that $G$ is bounded.
Then it can be seen that $f$ is locally bounded.
Since $f$ is also lower semicontinuous it follows that $f$ is continuous.
Now, suppose that $f$ is continuous at $0$.
As we can see in the question
Unbounded subdifferential of a convex functional,
this implies that $\partial f(0)$ is bounded.
Now we claim that $G\subset \partial f(0)$ holds.
Indeed, for each $g^*\in G$ we have
$ f(x)-f(0) = f(x) \geq \langle g^*, x \rangle $
and therefore $g^*\in\partial f(0)$.
Since $\partial f(0)$ is bounded this implies that $G$ has to be bounded, too.


TITLE: What is the non-relativistic limit of the quantised electromagnetic field?
QUESTION [2 upvotes]: I’m not a physicist so this question may be naive ... For a real scalar field, quantisation yields the Klein-Gordon equation and the non-relativistic limit of this gives the Schrödinger field. What is the equivalent equation or field starting from the electromagnetic field? I.e. quantising the EM field (E- and B-fields) and dispensing with Lorentz covariance gives what kind of equation or field? I.e. what’s the equivalent of the Schrödinger equation/field for an EM field (rather than a scalar field).
I have an application that observes the structure of the EM field but not Lorentz covariance.

REPLY [0 votes]: First we have to distinguish between first quantisation and second quantisation. In the first quantisation physical observables like momentum, position and energy become operators, mostly differential operators.
However, in many occasions physicists will consider the Klein-Gordon equation as classical equation for a field $\psi$ that in the second quantisation gets the status of an operator. To reduce the possible confusion: We won't talk about second quantisation here. In the following we will only talk about the "first" quantisation.
What happens in the first quantisation? Generally it consists of considering an energy-momentum relation for a particle as a dispersion-relation of a wave.
So in non-relativistic mechanics the energy momentum-relation of a particle in a potential $U$ is:
$$E = \frac{p^2}{2m} + U$$
We know in particular from the Davisson-Germer experiment that particles can also behave as waves. In order to find the wave equation that belongs to 1-particle motion one considers the energy momentum-relation as dispersion relation of the wave with wave vector $k$:
$$E = \frac{\hbar^2 k^2}{2m} + U$$
be replacing energy by $i\hbar\frac{\partial}{\partial t}$ and the wave vector by $-i\frac{\partial}{\partial x}$ we immediately get the Schroedinger-equation.
$$ i\hbar \frac{\partial}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial^2 x}  + U$$
The replacement of energy and wave vector by differential operators can be considered as the first-quantisation process. Because now we deal with (differential) operators for the description of the physical quantities.
For the Klein-Gordon-equation is the same. However, one now considers the relativistic energy-momentum-relation (without potential)of a particle:
$$E^2 = (pc)^2 + (mc^2)^2$$
Replacing  energy by $i\hbar\frac{\partial}{\partial t}$ and momentum by $-i\hbar\frac{\partial}{\partial x}$ we get the Klein-Gordon-equation:
$$\frac{1}{c^2}\frac{\partial^2}{\partial^2 t} = \frac{\partial^2}{\partial^2 x} - \frac{m^2 c^2}{\hbar^2} \quad \longrightarrow \frac{1}{c^2}\frac{\partial^2\psi}{\partial^2 t} - \frac{\partial^2\psi}{\partial^2 x} +\frac{m^2 c^2}{\hbar^2}\psi=0  $$
Of course, one lets operate the operators on some wave field or in non-relativistic physics on a wave function $\psi$ that propagates according to the corresponding dispersion relation.
How about the electromagnetic field (EM) ? Actually, in order to make it understandable one only has to put in the right context. Actually, the quantisation process here goes in the other direction, since both precedent cases considered a particle motion which was translated into a wave-motion. However, for the EM-field it is the other way around. One already has a wave-motion (at least this is the way one first gets to know the electromagnetic field) and associates a particle motion to it. This step is formally realised in the second quantisation but because  here we talk about first quantisation it is out of scope. But in order to make the analogy to the two precedent cases perfect we can start from the particle picture of the EM-field, the photons, and get back the corresponding wave equation in the same way as in the two precedent cases. The energy-momentum relation of a photon is:
$$ E=pc \quad\quad \text{or better in this context} \quad\quad E^2 = (pc)^2$$
This is actually a special case of the Klein-Gordon-equation with mass $m=0$.
Therefore by considering the energy-momentum relation as a dispersion relation of a wave followed by the replacement of energy and momentum by the corresponding differential operators ($E = i\hbar \frac{\partial }{\partial t}$ and $p= -i\hbar \frac{\partial}{\partial x}$) we get as wave-equation for the electromagnetic field:
$$\frac{1}{c^2}\frac{\partial^2}{\partial^2 t} = \frac{\partial^2}{\partial^2 x}$$
The is a well-known relation, it actually applies to the for components of the 4-vector potential:
$$\frac{1}{c^2}\frac{\partial^2 A^\mu}{\partial^2 t} = \frac{\partial^2 A^\mu}{\partial^2 x}$$
So we could actually conclude that the wave-function of the photon is the vector potential. Attention, the equation-analogy is one thing, the interpretations of these equations another. Yes one could say to some extent the vector potential is the wave-function of the photon. However, it does not have all the properties of the wave-function that solves the Schroedinger-equation. This is mainly due to the relativistic aspect of the wave-equation of the photon, details are here out of scope.
By the way, the wave-equation of the photon is perfectly Lorentz-covariant.
One might wonder about the right number of degrees of freedom of the photon field, this question is here also out of scope (see gauge freedom).


TITLE: Find the locus of intersection of tangents to an ellipse if the lines joining the points of contact to the centre be perpendicular.
QUESTION [1 upvotes]: Find the locus of intersection of tangents to an ellipse if the lines joining the points of contact to the centre be perpendicular.
Let the equation to the tangent be $$y=mx+\sqrt{a^2m^2+b^2} $$ This has to roots for $m$ that i.e $m_1$ and $m_2$.
perpendicular to this line passing through $(0,0)$ is $$my+x=0 $$ slope is $\frac{-1}{m}$
so for perpendiculars $$\frac{(-1)(-1)}{m_1m_2} =-1 $$ so $m_1m_2=-1 $
from equation of tangent $y=mx+\sqrt{a^2m^2+b^2} $
$$m^2(x^2-a^2)-2mxy+y^2-b^2=0 $$and hence locus is $$\frac{y^2-b^2}{x^2-a^2}=-1 $$ But this is not the correct answer. The answer is $b^4x^2+a^4y^2=a^2b^2(a^2+b^2) $. What's the error ?

REPLY [2 votes]: Let $(x_1,y_1)$ be the generic point of the locus.
It is well known that $$\frac {xx_1}{a^2}+\frac {yy_1}{b^2}=1$$ represents the line passing through the points of contact of the tangents from $(x_1,y_1)$.
Now consider the equation $$\frac {x^2}{a^2}+\frac {y^2}{b^2}-\left(\frac {xx_1}{a^2}+\frac {yy_1}{b^2}\right)^2=0$$ It is satisfied by the coordinates of the center and the points of contact.
Since it can be written $$x^2 \left(\frac {x_1^2}{a^4}-\frac 1{a^2}\right) + y^2 \left(\frac {y_1^2}{b^4}-\frac 1{b^2}\right) + \frac {2\,x\,y\,x_1y_1}{a^2b^2}=0$$
it is quadratic homogeneous so represents a pair of lines (degenerate conic)
, clearly the lines joining the points of contact to the centre.
It is not difficult to prove that the lines are mutually perpendicular iff the sum of the coefficients of $x^2$ and $y^2$ is zero, that is $$\frac {x_1^2}{a^4}+\frac {y_1^2}{b^4}=\frac 1{a^2}+\frac 1{b^2}$$


TITLE: Application of dominated convergence theorem
QUESTION [2 upvotes]: Suppose $\{f_n\}$ is a sequence of functions in $L^2[0,1]$ such that $f_n(x)\rightarrow 0$ almost everywhere on $[0,1]$. If $\|f_n(x)\|_{L^2[0,1]}\le 1$ for all $n$ , then $$\lim_{n\rightarrow \infty} \int_0^1 f_n(x)g(x)~dx =0$$ for all $g\in L^2[0,1]$.
I feel I have to use dominated convergence theorem, but I can't fined the dominating functions.  Thanks for helping.

REPLY [3 votes]: Here's an outline:
Fix $g\in L^2$.
Choose $\delta>0$ so that $\Bigl(\int_E|g|^2\Bigr)^{1/2}$ is small whenever $\mu(E)<\delta$.
By Egoroff, find a set $E$ of measure less than $\delta$ so that $f_n$ converges uniformly to $0$ off $E$. Choose $N$ so that for $n>N$, $|f_n|$ is small on $E^C$.
Then write:
$$
\Bigl| \int f_n g\, \Bigr|\le \int |f_n ||g| =\int_E |f_n ||g|+\int_{E^C}|f_n ||g|
$$
and apply Hölder to both integrals on the right.


TITLE: 2 Possible Generalizations of Cayley's Theorem?
QUESTION [3 upvotes]: I'm wondering about the following 2 generalizations of Cayley's Theorem (every group embeds in a symmetric group). If these are known to be true/false, references would be appreciated.
1) (Weak Version) Given any finite collection of (not necessarily distinct) finite groups, can we embed them simultaneously in a (finite) group so that they have pairwise-disjoint intersection (i.e., intersect only at the identity)?
2) (Strong Version) Given an arbitrary set of arbitrary groups, can we embed them simultaneously in some group so that they have pairwise-disjoint intersection? It seems like this version may run into set-theoretic difficulties; if so an explanation/reference for those would also be welcome.
Note of course that if such a group exists for a given collection, then we can embed it in a symmetric group by Cayley's Theorem (so these are indeed generalizations).
Edit: To make it more interesting / rule out the obvious answer pointed out in the comments, can we arrange the embeddings so that the groups' normalizers are pairwise disjoint?

REPLY [3 votes]: So the question now is, given two nontrivial finite groups $G,H$, can we embed them both in a finite group $X$ such that the normalizers of $G$ and $H$ in $X$ intersect trivially?
I think we can do that as follows. Suppose that we can find a module $V$ for $G \times H$ over a finite field, such that neither $G$ nor $H$ fix any nonzero vectors $v \in V$, and such that there exists a vector $v \in V$ such that no nontrivial element of $G \times H$ stabilizes $v$. Then we let $X$ be the semidirect product of $V$ by $G \times H$. Because of the first property, the normalizers of $G$ and $H$  in $X$ have trivial intersection with $V$, so they are both equal to $G \times H$. The second property ensures that the complements $G \times H$ and $(G \times H)^v$ intersect trivially. So the subgroups $G$ and $H^v$ have the desired property, because their normalizers in $X$ are $G \times H$ and $(G \times H)^v$.
We can contruct such a module $V$ as follows. Choose a prime $p$ not dividing $|G|$ or $|H|$ and let $V_1$ and $V_2$ be the deleted permutation modules for the regular permutation representations of $G$ and $H$ over ${\mathbb F}_p$. (So $V_1$ and $V_2$ have dimensions $|G|-1$ and $|H|-1$.) These have vectors $v_1$ and $v_2$ that are not stabilised by any nontrivial element of $G$ and $H$, respectively.  Now let $V$ be the $G \times H$ module $V_1 \otimes V_2$ and let $v =v_1 \otimes v_2$.


TITLE: De moivre's theorem based problem
QUESTION [0 upvotes]: How to solve this type of problems
As $(cosx + i sinx)^n =cos(nx) + i sin(nx)$ if n is an integer or rational.
If power is irrational then how to simplify.
Also we know that $i = cos \frac {π}{2} + i sin \frac {π}{2}  $.
So how to use it.
(https://i.stack.imgur.com/JAJts.jpg)

REPLY [0 votes]: If you are acquainted with few definition or properties of complex exponential and logarithms the given expression simplifies to $\large i^i$ the modulus of such is 1.


TITLE: Why $\langle I, J\rangle =R$ for distinct prime ideals $I$, $J$ of a principal ideal domain $R$?
QUESTION [1 upvotes]: Let $R$ be a principal ideal domain with identity and $I$, $J$ be distinct prime ideals of $R$. Prove that $1 \in \langle I, J\rangle$ hence $\langle I, J\rangle = R$. How to prove?

REPLY [0 votes]: Let $I=\langle p\rangle$ and $J=\langle q\rangle$ be two distinct prime ideals in the PID $R$. Therefore neither $p\mid q$ nor $q\mid p$.
Consider the ideal $\langle I,J\rangle=\langle c\rangle$. Since $p,q\in\langle c\rangle$ we have $c\mid p,q$, and since primes are irreducible, then either $c$ is $p$ times a unit (which is impossible since it would imply $p\mid q$), or $q$ times a unit (which is also impossible) or a unit itself, and therefore $\langle c\rangle=R$.