text:
\begin{document}

\begin{abstract}
	We first construct a closed spherical CR manifold of dimension at least five
	with non-trivial first Chern class in real coefficients.
	In particular,
	it admits no pseudo-Einstein contact forms.
	We next give a constraint on Chern classes
	of (not necessarily closed) spherical CR manifolds in real coefficients.
	Finally we obtain a topological obstruction to the existence of a spherical CR structure
	on a given co-oriented contact manifold.
\end{abstract}

\maketitle

\section{Introduction}
\label{section:introduction}

In conformal geometry,
locally conformally flat manifolds,
or $(PO(n+1, 1), S^{n})$-manifolds,
are an important class of conformal manifolds.
Chern and Simons~\cite{Chern-Simons1971} and Avez~\cite{Avez1970}
have proved that the Pontrjagin classes of any locally conformally flat manifold are trivial in real coefficients.
In this paper,
we consider a similar problem for ``locally CR flat'' manifolds,
which are known as \emph{spherical CR manifolds},
or $(PU(n+1, 1), S^{2n+1})$-manifolds.

We will first construct a closed spherical CR manifolds of dimension at least five
with non-trivial first Chern class in real coefficients.
In particular,
the vanishing of characteristic classes does not occur for spherical CR manifolds in general
as opposed to locally conformally flat manifolds.

\begin{theorem}
\label{thm:spherical-CR-with-non-trivial-first-Chern-class}
	For each integer $n \geq 2$,
	there exists a closed spherical CR manifold $(S, T^{1,0}S)$ of dimension $2n+1$
	with $c_{1}(T^{1,0}S) \neq 0$ in $H^{2}(S, \bbR)$.
	In particular,
	$S$ has no pseudo-Einstein contact forms.
\end{theorem}

Note that there exist no known examples of spherical CR three-manifolds
with non-trivial first Chern class in real coefficients.

However,
we will obtain a constraint on higher Chern classes of spherical CR manifolds
in terms of the first Chern class.

\begin{theorem}
\label{thm:Chern-classes-of-spherical-CR-structure}
	Let $(M, T^{1,0}M)$ be a spherical CR manifold of dimension $2n+1$.
	Then
	\begin{equation}
		c_{k}(T^{1,0}M)
		= \frac{1}{(n+2)^{k}} \binom{n+2}{k} c_{1}(T^{1,0}M)^{k}
	\end{equation}
	in $H^{2k}(M, \bbR)$.
	In particular,
	\begin{equation}
		c_{2}(T^{1,0}M)
		= \frac{n+1}{2(n+2)} c_{1}(T^{1,0}M)^{2}
	\end{equation}
	in $H^{4}(M, \bbR)$.
\end{theorem}

The following proposition implies that
it is essential to consider the cohomology in real coefficients.

\begin{proposition}
\label{prop:counterexample-to-integer-coefficients}
	For each integer $n \geq 2$,
	there exists a closed spherical CR manifold $(S, T^{1,0}S)$ of dimension $2n+1$
	such that
	\begin{equation}
		2(n+2) c_{2}(T^{1,0}S)
		\neq (n+1) c_{1}(T^{1,0}S)^{2}
	\end{equation}
	in $H^{4}(S, \bbZ)$.
\end{proposition}

A spherical CR structure $T^{1,0}M$ has the underlying co-oriented contact structure $\xi \coloneqq \Re T^{1,0}M$.
Conversely,
it is natural to ask whether a given co-oriented contact structure $\xi$ admits a spherical CR structure.
\cref{thm:Chern-classes-of-spherical-CR-structure} gives a topological obstruction to this problem.

\begin{theorem}
\label{thm:topological-obstruction-to-spherical-CR-structure}
	Let $(M, \xi)$ be a $(2n+1)$-dimensional co-oriented contact manifold.
	If $\xi$ admits a spherical CR structure,
	then
	\begin{equation}
		c_{k}(\xi)
		= \frac{1}{(n+2)^{k}} \binom{n+2}{k} c_{1}(\xi)^{k}
	\end{equation}
	in $H^{2k}(M, \bbR)$.
	In particular,
	\begin{equation}
	\label{eq:second-Chern-contact-with-spherical}
		c_{2}(\xi)
		= \frac{n+1}{2(n+2)} c_{1}(\xi)^{2}
	\end{equation}
	in $H^{4}(M, \bbR)$.
\end{theorem}

This theorem follows from \cref{thm:Chern-classes-of-spherical-CR-structure}
and the fact that $c(\xi) = c(T^{1,0}M)$.
We apply this for a Stein fillable contact manifold.

\begin{proposition}
\label{prop:Stein-fillable-with-no-spherical-CR}
	For each integer $n \geq 3$,
	there exists a Stein fillable contact manifold $(M, \xi)$ of dimension $2n+1$ such that
	\begin{equation}
		c_{2}(\xi)
		\neq \frac{n+1}{2(n+2)} c_{1}(\xi)^{2}
	\end{equation}
	in $H^{4}(M, \bbR)$.
	In particular,
	$\xi$ admits no spherical CR structures.
\end{proposition}

Note that there exist no Stein fillable contact manifolds of dimension five violating \cref{eq:second-Chern-contact-with-spherical}.
This is because both the second Chern class and the square of the first Chern class must vanish in real coefficients
on any five-dimensional Stein fillable (more generally, holomorphically fillable) contact manifold;
see~\cite{Takeuchi2020} for example.

This paper is organized as follows.
In \cref{section:CR-geometry},
we recall basic facts on CR and Sasakian manifolds
and give a proof of \cref{thm:spherical-CR-with-non-trivial-first-Chern-class}.
\cref{section:proof-of-main-theorem} is devoted to a proof of \cref{thm:Chern-classes-of-spherical-CR-structure}.
In \cref{section:related-examples},
we construct some examples related to \cref{thm:Chern-classes-of-spherical-CR-structure,thm:topological-obstruction-to-spherical-CR-structure}.

\medskip

\noindent
\emph{Notation.}
We use Einstein's summation convention
and assume that lowercase Greek indices $\alpha, \beta, \gamma, \dots$ run from $1, \dots, n$.

\medskip

\section{CR geometry}
\label{section:CR-geometry}

\subsection{CR structures}
\label{subsection:CR-structures}

Let $M$ be an orientable smooth $(2n+1)$-dimensional manifold without boundary.
A \emph{CR structure} is a rank $n$ complex subbundle $T^{1,0}M$
of the complexified tangent bundle $TM \otimes \mathbb{C}$ such that
\begin{equation}
	T^{1,0}M \cap T^{0,1}M = 0, \qquad
	[\Gamma(T^{1,0}M), \Gamma(T^{1,0}M)] \subset \Gamma(T^{1,0}M),
\end{equation}
where $T^{0,1}M$ is the complex conjugate of $T^{1,0}M$ in $TM \otimes \bbC$.
A typical example of CR manifolds is a real hypersurface $M$ in an $(n+1)$-dimensional complex manifold $X$;
this $M$ has the canonical CR structure
\begin{equation}
	T^{1,0}M
	\coloneqq T^{1,0}X |_{M} \cap (TM \otimes \bbC).
\end{equation}
In particular,
the unit sphere $S^{2n+1}$ in $\bbC^{n+1}$ has the canonical CR structure $T^{1,0}S^{2n+1}$.
A CR manifold $(M, T^{1,0}M)$ is said to be \emph{spherical}
if it is locally isomorphic to $(S^{2n+1}, T^{1,0} S^{2n+1})$.
Denote by $K_{M}$ the subbundle of $\bigwedge\nolimits^{n + 1} (T^{\ast}M \otimes \bbC)$
defined by the equation $\ovZ \contr \zeta = 0$ for all $Z \in T^{1,0}M$,
called the \emph{canonical bundle} of $M$.
For a real number $w$,
we define the bundle $\calE(w)$ of \emph{$w$-densities} by
\begin{equation}
	\calE(w)
	\coloneqq (K_{M} \otimes \overline{K_{M}})^{-w/(n+2)}.
\end{equation}
(If $K_{M} \setminus \{0\}$ is viewed as a principal $\bbC^{\times}$-bundle,
then $\calE(w)$ is the complex line bundle
induced by the representation $\lambda \mapsto \abs{\lambda}^{-2w/(n+2)}$.)

A CR structure $T^{1,0}M$ is said to be \emph{strictly pseudoconvex}
if there exists a nowhere-vanishing real one-form $\theta$ on $M$
such that
$\theta$ annihilates $T^{1,0}M$ and
\begin{equation}
	-\sqrt{- 1} d \theta (Z, \ovZ) > 0, \qquad
	0 \neq Z \in T^{1,0}M.
\end{equation}
We call such a one-form a \emph{contact form}.
The triple $(M, T^{1,0}M, \theta)$ is called a \emph{pseudo-Hermitian manifold}.
Denote by $T$ the \emph{Reeb vector field} with respect to $\theta$;
that is,
the unique vector field satisfying
\begin{equation}
	\theta(T) = 1, \qquad T \contr d\theta = 0.
\end{equation}
Let $(\tensor{Z}{_{\alpha}})$ be a local frame of $T^{1,0}M$,
and set $\tensor{Z}{_{\ovxa}} = \overline{\tensor{Z}{_{\alpha}}}$.
Then
$(T, \tensor{Z}{_{\alpha}}, \tensor{Z}{_{\ovxa}})$ gives a local frame of $TM \otimes \mathbb{C}$,
called an \emph{admissible frame}.
Its dual frame $(\theta, \tensor{\theta}{^{\alpha}}, \tensor{\theta}{^{\ovxa}})$
is called an \emph{admissible coframe}.
The two-form $d \theta$ is written as
\begin{equation}
	d \theta
	= \sqrt{- 1} \tensor{l}{_{\alpha}_{\ovxb}} \tensor{\theta}{^{\alpha}} \wedge \tensor{\theta}{^{\ovxb}},
\end{equation}
where $(\tensor{l}{_{\alpha}_{\ovxb}})$ is a positive definite Hermitian matrix.
We use $\tensor{l}{_{\alpha}_{\ovxb}}$ and its inverse $\tensor{l}{^{\alpha}^{\ovxb}}$
to raise and lower indices of tensors.

Let $\zeta$ be a (locally defined) nowhere-vanishing section of $K_{M}$.
We denote by $\abs{\zeta}^{2w}$ the section $(\zeta \otimes \ovxz)^{w}$ of $\calE(-(n+2)w)$.
A contact form $\theta$ is \emph{volume normalized} by $\zeta$ if
\begin{equation}
	\theta \wedge (d \theta)^{n}
	= (\sqrt{- 1})^{n^{2}}(n !) \theta \wedge (T \contr \zeta) \wedge (T \contr \ovxz).
\end{equation}
Conversely,
for a contact form $\theta$,
there exists a (locally defined) nowhere-vanishing section $\zeta$ of $K_{M}$
such that $\theta$ is volume normalized by $\zeta$.
Such a $\zeta$ is determined uniquely by $\theta$ modulo multiplication by $S^{1}$-valued smooth functions.
Hence $\abs{\zeta}^{2w}$ is independent of the choice of $\zeta$
and defines a nowhere-vanishing global section of $\calE(-(n+2)w)$.
In particular,
$\calE(w)$ is a trivial complex line bundle.

An important example of strictly pseudoconvex CR manifolds is the boundary of a strictly pseudoconvex domain.
Let $\Omega$ be a relatively compact domain in an $(n+1)$-dimensional complex manifold $X$
with smooth boundary $M = \partial \Omega$.
Then there exists a smooth function $\rho$ on $X$ such that
\begin{equation}
	\Omega = \rho^{-1}((-\infty, 0)), \quad
	M = \rho^{-1}(0), \quad
	d \rho \neq 0 \ \text{on} \ M;
\end{equation}
such a $\rho$ is called a \emph{defining function} of $\Omega$.
A domain $\Omega$ is said to be \emph{strictly pseudoconvex}
if we can take a defining function $\rho$ of $\Omega$ that is strictly plurisubharmonic near $M$.
Then $M$ is a closed strictly pseudoconvex real hypersurface
and $d^{c} \rho |_{M}$ is a contact form on $M$,
where $d^{c} \coloneqq (\sqrt{-1}/2) (\delb - \del)$.
In particular,
$(S^{2n+1}, T^{1,0}S^{2n+1})$ is a strictly pseudoconvex CR manifold;
this is because $S^{2n+1}$ is the boundary of the unit ball in $\bbC^{n+1}$,
which is a strictly pseudoconvex domain.
This implies that any spherical CR manifold is strictly pseudoconvex.

We call $\Omega$ a \emph{Stein domain}
if $\Omega$ admits a defining function $\rho$
that is strictly plurisubharmonic on a neighborhood of the closure of $\Omega$.
Note that a Stein domain is a Stein manifold;
this is because $-1 / \rho$ defines a strictly plurisubharmonic exhaustion function on $\Omega$.
A co-oriented contact structure $\xi$ on a closed $(2n+1)$-dimensional manifold $M$
is \emph{Stein fillable} if  there exists a Stein domain $\Omega$ of dimension $n+1$
such that $(\bdry \Omega, \Re T^{1,0}\bdry \Omega)$ is contactmorphic to $(M, \xi)$.

\subsection{Tanaka-Webster connection}
\label{subsection:TW-connection}

A contact form $\theta$ induces a canonical affine connection $\nabla$,
called the \emph{Tanaka-Webster connection} with respect to $\theta$.
It is defined by
\begin{equation}
	\nabla T
	= 0,
	\quad
	\nabla \tensor{Z}{_{\alpha}}
	= \tensor{\omega}{_{\alpha}^{\beta}} \tensor{Z}{_{\beta}},
	\quad
	\nabla \tensor{Z}{_{\ovxa}}
	= \tensor{\omega}{_{\ovxa}^{\ovxb}} \tensor{Z}{_{\ovxb}}
	\quad
	\pqty{ \tensor{\omega}{_{\ovxa}^{\ovxb}}
	\coloneqq \overline{\tensor{\omega}{_{\alpha}^{\beta}}} }
\end{equation}
with the following structure equations:
\begin{gather}
\label{eq:str-eq-of-TW-conn1}
	d \tensor{\theta}{^{\beta}}
	= \tensor{\theta}{^{\alpha}} \wedge \tensor{\omega}{_{\alpha}^{\beta}}
	+ \tensor{A}{^{\beta}_{\ovxa}} \theta \wedge \tensor{\theta}{^{\ovxa}}, \\
\label{eq:str-eq-of-TW-conn2}
	d \tensor{l}{_{\alpha}_{\ovxb}}
	= \tensor{\omega}{_{\alpha}^{\gamma}} \tensor{l}{_{\gamma}_{\ovxb}}
	+ \tensor{l}{_{\alpha}_{\ovxg}} \tensor{\omega}{_{\ovxb}^{\ovxg}}.
\end{gather}
The tensor $\tensor{A}{_{\alpha}_{\beta}} \coloneqq \overline{\tensor{A}{_{\ovxa}_{\ovxb}}}$
is shown to be symmetric and called the \emph{Tanaka-Webster torsion}.

The curvature form
$\tensor{\Omega}{_{\alpha}^{\beta}} \coloneqq d \tensor{\omega}{_{\alpha}^{\beta}}
- \tensor{\omega}{_{\alpha}^{\gamma}} \wedge \tensor{\omega}{_{\gamma}^{\beta}}$
of the Tanaka-Webster connection satisfies
\begin{equation}
\label{eq:curvature-form-of-TW-connection}
	\begin{split}
		\tensor{\Omega}{_{\alpha}^{\beta}}
		&= \tensor{R}{_{\alpha}^{\beta}_{\gamma}_{\ovxs}} \tensor{\theta}{^{\gamma}} \wedge \tensor{\theta}{^{\ovxs}}
			- \tensor{\nabla}{^{\beta}} \tensor{A}{_{\alpha}_{\gamma}} \theta \wedge \tensor{\theta}{^{\gamma}}
			+ \tensor{\nabla}{_{\alpha}} \tensor{A}{^{\beta}_{\ovxg}} \theta \wedge \tensor{\theta}{^{\ovxg}}  \\
		&\quad - \sqrt{-1} \tensor{A}{_{\alpha}_{\gamma}} \tensor{\theta}{^{\gamma}} \wedge \tensor{\theta}{^{\beta}}
			+ \sqrt{-1} \tensor{l}{_{\alpha}_{\ovxg}} \tensor{A}{^{\beta}_{\ovxr}}
			\tensor{\theta}{^{\ovxg}} \wedge \tensor{\theta}{^{\ovxr}}.
	\end{split}
\end{equation}
We call the tensor $\tensor{R}{_{\alpha}^{\beta}_{\gamma}_{\ovxs}}$
the \emph{Tanaka-Webster curvature}.
This tensor has the symmetry
\begin{equation}
	\tensor{R}{_{\alpha}_{\ovxb}_{\gamma}_{\ovxs}}
	= \tensor{R}{_{\gamma}_{\ovxb}_{\alpha}_{\ovxs}}
	= \tensor{R}{_{\alpha}_{\ovxs}_{\gamma}_{\ovxb}}.
\end{equation}
Contraction of indices gives the \emph{Tanaka-Webster Ricci curvature}
$\tensor{\Ric}{_{\gamma}_{\ovxs}} = \tensor{R}{_{\alpha}^{\alpha}_{\gamma}_{\ovxs}}$
and the \emph{Tanaka-Webster scalar curvature}
$\Scal = \tensor{\Ric}{_{\gamma}^{\gamma}}$.
The \emph{Chern tensor} $\tensor{S}{_{\alpha}_{\ovxb}_{\gamma}_{\ovxs}}$ is defined by
\begin{equation}
	\tensor{S}{_{\alpha}_{\ovxb}_{\gamma}_{\ovxs}}
	\coloneqq \tensor{R}{_{\alpha}_{\ovxb}_{\gamma}_{\ovxs}}
		- \tensor{P}{_{\alpha}_{\ovxb}} \tensor{l}{_{\gamma}_{\ovxs}}
		- \tensor{P}{_{\gamma}_{\ovxb}} \tensor{l}{_{\alpha}_{\ovxs}}
		- \tensor{P}{_{\gamma}_{\ovxs}} \tensor{l}{_{\alpha}_{\ovxb}}
		- \tensor{P}{_{\alpha}_{\ovxs}} \tensor{l}{_{\gamma}_{\ovxb}},
\end{equation}
where
\begin{equation}
	\tensor{P}{_{\alpha}_{\ovxb}}
	\coloneqq \frac{1}{n+2} \pqty{\tensor{\Ric}{_{\alpha}_{\ovxb}} - \frac{\Scal}{2(n+1)} \tensor{l}{_{\alpha}_{\ovxb}}}.
\end{equation}
Note that the trace $P \coloneqq \tensor{P}{_{\alpha}^{\alpha}}$ is equal to $\Scal / 2(n+1)$.
It is known that $(M, T^{1,0}M)$ is a spherical CR manifold
if and only if the Chern tensor vanishes identically when $n \geq 2$~\cite{Chern-Moser1974}.
The tensor $\tensor{V}{_{\alpha}_{\ovxb}_{\gamma}}$ is defined by
\begin{equation}
	\tensor{V}{_{\alpha}_{\ovxb}_{\gamma}}
	\coloneqq \tensor{\nabla}{_{\ovxb}} \tensor{A}{_{\alpha}_{\gamma}}
		+ \sqrt{-1} \tensor{\nabla}{_{\gamma}} \tensor{P}{_{\alpha}_{\ovxb}}
		- \sqrt{-1} \tensor{T}{_{\gamma}} \tensor{l}{_{\alpha}_{\ovxb}}
		- 2 \sqrt{-1} \tensor{T}{_{\alpha}} \tensor{l}{_{\gamma}_{\ovxb}},
\end{equation}
where
\begin{equation}
	\tensor{T}{_{\alpha}}
	\coloneqq\frac{1}{n+2} (\tensor{\nabla}{_{\alpha}} P - \sqrt{-1} \tensor{\nabla}{^{\beta}} \tensor{A}{_{\alpha}_{\beta}}).
\end{equation}
This tensor is a divergence of the Chern tensor:
\begin{equation}
\label{eq:divergence-of-Chern-tensor}
	\tensor{\nabla}{^{\ovxs}} \tensor{S}{_{\alpha}_{\ovxb}_{\gamma}_{\ovxs}}
	= - n \sqrt{-1} \tensor{V}{_{\alpha}_{\ovxb}_{\gamma}};
\end{equation}
see~\cite{Case-Gover2017-preprint}*{(2.5)}.

Assume that $n \geq 2$.
A contact form $\theta$ is said to be \emph{pseudo-Einstein} if
\begin{equation}
\label{eq:pseudo-Einstein-condition}
	\tensor{\Ric}{_{\alpha}_{\overline{\beta}}}
	= \frac{1}{n} \Scal \cdot \tensor{l}{_{\alpha}_{\overline{\beta}}}.
\end{equation}
It is known that $c_{1}(T^{1,0}M) = 0$ in $H^{2}(M, \bbR)$
if $(M, T^{1,0}M)$ has a pseudo-Einstein contact form~\cite{Lee1988}*{Proposition D}.

\subsection{Sasakian manifolds}
\label{subsection:Sasakian-manifolds}

Sasakian manifolds are an important class of pseudo-Hermitian manifolds.
See~\cite{Boyer-Galicki2008} for a comprehensive introduction to Sasakian manifolds.

A \emph{Sasakian manifold} is a pseudo-Hermitian manifold $(S, T^{1,0}S, \eta)$
with vanishing Tanaka-Webster torsion.
This condition is equivalent to that the Reeb vector field $T$ with respect to $\eta$
preserves the CR structure $T^{1,0}S$.

A typical example of Sasakian manifolds
is the circle bundle associated with a negative Hermitian line bundle.
Let $Y$ be an $n$-dimensional complex manifold
and $(L, h)$ a Hermitian holomorphic line bundle over $Y$
such that
\begin{equation}
	\omega
	\coloneqq - \sqrt{-1} \Theta_{h} = d d^{c} \log h
\end{equation}
is a \Kahler form on $Y$.
Consider the circle bundle
\begin{equation}
	S
	\coloneqq \Set{ v \in L | h(v, v) = 1 }
\end{equation}
over $Y$,
which is a real hypersurface in the total space of $L$.
The one-form $\eta \coloneqq d^{c} \log h |_{S}$
is a connection one-form of the principal $S^{1}$-bundle $p \colon S \to Y$
and satisfies $d \eta = p^{\ast} \omega$.
Moreover,
the natural CR structure $T^{1,0}S$ coincides with the horizontal lift of $T^{1,0}Y$ with respect to $\eta$.
Since $\omega$ is a \Kahler form,
we have
\begin{equation}
	- \sqrt{-1} d \eta (Z, \ovZ)
	= - \sqrt{-1} \omega (p_{\ast} Z, p_{\ast} \ovZ)
	> 0
\end{equation}
for all non-zero $Z \in T^{1,0}S$.
This implies that $(S, T^{1,0}S)$ is a strictly pseudoconvex CR manifold
and $\eta$ is a contact form on $S$.
We call this pseudo-Hermitian manifold $(S, T^{1,0}S, \eta)$
the \emph{circle bundle associated with $(Y, L, h)$}.

Consider the Tanaka-Webster connection with respect to $\eta$.
Take a local coordinate $(z^{1}, \dots , z^{n})$ of $Y$.
The \Kahler form $\omega$ is written as
\begin{equation}
	\omega
	= \sqrt{-1} \tensor{g}{_{\alpha}_{\ovxb}} d z^{\alpha} \wedge d \ovz^{\beta},
\end{equation}
where $(\tensor{g}{_{\alpha}_{\ovxb}})$ is a positive definite Hermitian matrix.
Let $Z_{\alpha}$ be the horizontal lift of $\pdvf{}{z^{\alpha}}$.
Then $(T, Z_{\alpha}, Z_{\ovxa} \coloneqq \overline{Z_{\alpha}})$
is an admissible frame on $S$.
The corresponding admissible coframe is given by
$(\eta, \theta^{\alpha} \coloneqq p^{\ast} (d z^{\alpha}), \theta^{\ovxa} \coloneqq p^{\ast} (d \ovz^{\alpha}))$.
Since $d \eta = p^{*} \omega$,
we have
\begin{equation}
	d \eta = \sqrt{-1} (p^{\ast} \tensor{g}{_{\alpha}_{\ovxb}}) \theta^{\alpha} \wedge \theta^{\ovxb},
\end{equation}
which implies $\tensor{l}{_{\alpha}_{\ovxb}} = p^{\ast} \tensor{g}{_{\alpha}_{\ovxb}}$.
The connection form $\tensor{\pi}{_{\alpha}^{\beta}}$ of the \Kahler metric
with respect to the frame $(\pdvf{}{\tensor{z}{^{\alpha}}})$ satisfies
\begin{equation} \label{eq:structure-equation-for-Kahler-metric}
	0 = d (d z^{\beta}) = d \tensor{z}{^{\alpha}} \wedge \tensor{\pi}{_{\alpha}^{\beta}},
	\qquad
	d \tensor{g}{_{\alpha}_{\ovxb}}
	= \tensor{\pi}{_{\alpha}^{\gamma}} \tensor{g}{_{\gamma}_{\ovxb}}
	+ \tensor{g}{_{\alpha}_{\ovxg}} \tensor{\pi}{_{\ovxb}^{\ovxg}}
	\qquad
	\pqty{\tensor{\pi}{_{\ovxa}^{\ovxb}}
	\coloneqq \overline{\tensor{\pi}{_{\alpha}^{\beta}}}}.
\end{equation}
We write as $\tensor{\Pi}{_{\alpha}^{\beta}}$ the curvature form of the \Kahler metric.
Pulling back \cref{eq:structure-equation-for-Kahler-metric} by $p$ gives
\begin{equation}
	d \theta^{\beta} = \tensor{\theta}{^{\alpha}} \wedge (p^{*} \tensor{\pi}{_{\alpha}^{\beta}}),
	\qquad
	d \tensor{l}{_{\alpha}_{\ovxb}}
	= (p^{\ast} \tensor{\pi}{_{\alpha}^{\gamma}}) \tensor{l}{_{\gamma}_{\ovxb}}
	+ \tensor{l}{_{\alpha}_{\ovxg}} (p^{\ast} \tensor{\pi}{_{\ovxb}^{\ovxg}}).
\end{equation}
This yields that
$\tensor{\omega}{_{\alpha}^{\beta}} = p^{\ast} \tensor{\pi}{_{\alpha}^{\beta}}$,
and the Tanaka-Webster torsion vanishes identically;
that is,
$(S, T^{1,0}S, \eta)$ is a Sasakian manifold.
Moreover,
the curvature form $\tensor{\Omega}{_{\alpha}^{\beta}}$ of the Tanaka-Webster connection
is given by $\tensor{\Omega}{_{\alpha}^{\beta}} = p^{\ast} \tensor{\Pi}{_{\alpha}^{\beta}}$.
In particular,
$(S, T^{1, 0} S)$ is a spherical CR manifold
if and only if $\omega$ defines a Bochner-flat \Kahler metric on $Y$ when $n \geq 2$.

Now we construct an example of closed spherical CR manifolds of dimension at least five
with non-trivial first Chern class in real coefficients.

\begin{proof}[Proof of \cref{thm:spherical-CR-with-non-trivial-first-Chern-class}]
	Let $Y_{1}$ be a closed Riemann surface of genus two.
	Take a Hermitian metric $h_{1}$ on $L_{1} \coloneqq T^{1, 0} Y_{1}$
	such that $\omega_{1} \coloneqq - \sqrt{-1} \Theta_{h_{1}}$ defines a \Kahler-Einstein metric on $Y_{1}$
	with Einstein constant $-1$.
	Denote by $h_{2}$ a Hermitian metric on $L_{2} \coloneqq \calO(-2)$ over $Y_{2} \coloneqq \cps^{n-1}$
	such that $\omega_{2} \coloneqq - \sqrt{-1} \Theta_{h_{2}}$
	is a positive constant multiple of the Fubini-Study form.
	Consider the Hermitian holomorphic line bundle
	$(L \coloneqq L_{1} \boxtimes L_{2}, h \coloneqq h_{1} \boxtimes h_{2})$ over $Y \coloneqq Y_{1} \times Y_{2}$.
	Then $\omega \coloneqq - \sqrt{-1} \Theta_{h} = \omega_{1} + \omega_{2}$ defines
	a Bochner-flat \Kahler metric on $Y$~\cite{Tachibana-Liu1970}*{Section 2}.
	Let $(S, T^{1,0}S, \eta)$ be the circle bundle associated with $(Y, L, h)$.
	Since $\omega$ is Bochner-flat,
	$(S, T^{1,0}S)$ is a spherical CR manifold.
	It suffices to show $c_{1}(T^{1,0}S) \neq 0$ in $H^{2}(S, \bbR)$.
	The first Chern classes of $L$ and $T^{1,0}Y$ are given by
	\begin{gather}
		c_{1}(L)
		= c_{1}(L_{1}) + c_{1}(L_{2}), \\
		c_{1}(T^{1,0}Y)
		= c_{1}(T^{1,0}Y_{1}) + c_{1}(T^{1,0}Y_{2})
		= c_{1}(L_{1}) - \frac{n}{2} c_{1}(L_{2}).
	\end{gather}
	Consider the Gysin exact sequence
	\begin{equation}
		H^{0}(Y, \bbR) \cong \bbR \xrightarrow{c_{1}(L)}
		H^{2}(Y, \bbR) \xrightarrow{p^{\ast}}
		H^{2}(S, \bbR).
	\end{equation}
	Since $c_{1}(T^{1,0}Y)$ is not proportional to $c_{1}(L)$,
	the cohomology class $p^{\ast} c_{1}(T^{1,0}Y)$ is not equal to zero in $H^{2}(S, \bbR)$.
	From $p^{\ast} T^{1,0}Y \cong T^{1,0}S$,
	it follows that $c_{1}(T^{1,0}S) \neq 0$ in $H^{2}(S, \bbR)$.
	In particular,
	$(S, T^{1,0}S)$ has no pseudo-Einstein contact forms.
\end{proof}

\section{Proof of \cref{thm:Chern-classes-of-spherical-CR-structure}}
\label{section:proof-of-main-theorem}

Let $(M, T^{1,0}M)$ be a strictly pseudoconvex CR manifold of dimension $2 n + 1$.
Denote by $\calT$ the complex vector bundle $\calE(1) \oplus T^{1,0}M \oplus \calE(0)$ of rank $n+2$.
Since both $\calE(1)$ and $\calE(0)$ are trivial line bundles,
we have $c(T^{1,0}M) = c(\calT)$.
Hence it suffices to study $c(\calT)$ for a proof of \cref{thm:Chern-classes-of-spherical-CR-structure}.
In the proof of~\cite{Marugame2019-preprint}*{Proposition 5.4},
Marugame has introduced a connection $\nabla^{\calT}$ on $\calT$ via the CR tractor connection~\cite{Gover-Graham2005};
this $\calT$ coincides with $\tensor{\calE}{^{A}}(1,0)$ in~\cite{Marugame2019-preprint}.
The curvature form $\Omega^{\calT}$ of $\nabla^{\calT}$ satisfies
\begin{equation}
\label{eq:tractor-curvature}
	\Omega^{\calT}
	=
	\begin{pmatrix}
		0 & 0 & 0 \\
		\ast & \tensor{\Xi}{_{\alpha}^{\beta}} & 0 \\
		\ast & \ast & 0
	\end{pmatrix}
	+ \frac{1}{n + 2} \tensor{\Omega}{_{\gamma}^{\gamma}}
	\begin{pmatrix}
		1 & 0 & 0 \\
		0 & \tensor{\delta}{_{\alpha}^{\beta}} & 0 \\
		0 & 0 & 1
	\end{pmatrix}
	,
\end{equation}
where
\begin{equation}
\label{eq:trace-free-curvature}
	\tensor{\Xi}{_{\alpha}^{\beta}}
	\coloneqq \tensor{S}{_{\alpha}^{\beta}_{\gamma}_{\ovxs}} \tensor{\theta}{^{\gamma}} \wedge \tensor{\theta}{^{\ovxs}}
		- \tensor{V}{_{\alpha}^{\beta}_{\gamma}} \theta \wedge \tensor{\theta}{^{\gamma}}
		+ \tensor{V}{^{\beta}_{\alpha}_{\ovxg}} \theta \wedge \tensor{\theta}{^{\ovxg}}.
\end{equation}

\begin{proof}[Proof of \cref{thm:Chern-classes-of-spherical-CR-structure}]
	Since $(M, T^{1,0}M)$ is spherical,
	the Chern tensor $\tensor{S}{_{\alpha}^{\beta}_{\gamma}_{\ovxs}}$ vanishes identically.
	It follows from \cref{eq:divergence-of-Chern-tensor}
	that the same is true for $\tensor{V}{_{\alpha}^{\beta}_{\gamma}}$.
	Thus we have $\tensor{\Xi}{_{\alpha}^{\beta}} = 0$.
	Combining this with \cref{eq:tractor-curvature} yields that
	\begin{equation}
		\det(\id_{\calT} + \frac{\sqrt{- 1}}{2 \pi} \Omega^{\calT})
		= \pqty{ 1 + \frac{1}{n+2} \frac{\sqrt{-1}}{2 \pi} \tensor{\Omega}{_{\gamma}^{\gamma}} }^{n+2}
	\end{equation}
	is a representative of $c(\calT) = c(T^{1,0}M)$.
	Since $(\sqrt{-1} / 2 \pi) \tensor{\Omega}{_{\gamma}^{\gamma}}$ is a representative of $c_{1}(T^{1,0}M)$,
	we have the desired conclusion.
\end{proof}

\section{Related examples}
\label{section:related-examples}

We first construct a spherical CR manifold with non-trivial second Chern class.

\begin{proposition}
\label{prop:spherical-CR-with-non-trivial-second-Chern-class}
	For each integer $n \geq 4$,
	there exists a closed spherical CR manifold $(S, T^{1,0}S)$ of dimension $2n+1$ with
	\begin{equation}
		c_{2}(T^{1,0}S)
		= \frac{n+1}{2(n+2)} c_{1}(T^{1,0}S)^{2}
		\neq 0
	\end{equation}
	in $H^{4}(S, \bbR)$.
\end{proposition}

\begin{proof}
	Let $Y_{1}$ be a fake projective plane;
	that is,
	a closed complex hyperbolic manifold with the same Betti numbers as $\cps^{2}$;
	see~\cite{Mumford1979} for example.
	Take a Hermitian metric $h_{1}$ on $L_{1} \coloneqq K_{Y_{1}}^{-1}$
	such that $\omega_{1} \coloneqq - \sqrt{-1} \Theta_{h_{1}}$ defines a \Kahler metric on $Y_{1}$
	with negative constant holomorphic sectional curvature.
	Denote by $h_{2}$ a Hermitian metric on $L_{2} \coloneqq \calO(-3)$ over $Y_{2} \coloneqq \cps^{n-2}$
	such that $\omega_{2} \coloneqq - \sqrt{-1} \Theta_{h_{2}}$
	is a positive constant multiple of the Fubini-Study form.
	Consider the Hermitian holomorphic line bundle
	$(L \coloneqq L_{1} \boxtimes L_{2}, h \coloneqq h_{1} \boxtimes h_{2})$ over $Y \coloneqq Y_{1} \times Y_{2}$.
	Then $\omega \coloneqq - \sqrt{-1} \Theta_{h} = \omega_{1} + \omega_{2}$ defines
	a Bochner-flat \Kahler metric on $Y$~\cite{Tachibana-Liu1970}*{Section 2}.
	Let $(S, T^{1,0}S, \eta)$ be the circle bundle associated with $(Y, L, h)$.
	Since $\omega$ is Bochner-flat,
	$(S, T^{1,0}S)$ is a spherical CR manifold.
	It suffices to show $c_{1}(T^{1,0}S)^{2} \neq 0$ in $H^{4}(S, \bbR)$
	by \cref{thm:Chern-classes-of-spherical-CR-structure}.
	The first Chern classes of $L$ and $T^{1,0}Y$ are given by
	\begin{gather}
		c_{1}(L)
		= c_{1}(L_{1}) + c_{1}(L_{2}), \\
		c_{1}(T^{1,0}Y)
		= c_{1}(T^{1,0}Y_{1}) + c_{1}(T^{1,0}Y_{2})
		= c_{1}(L_{1}) - \frac{n-1}{3} c_{1}(L_{2}).
	\end{gather}
	Consider the Gysin exact sequence
	\begin{equation}
		H^{2}(Y, \bbR) \xrightarrow{c_{1}(L)}
		H^{4}(Y, \bbR) \xrightarrow{p^{\ast}}
		H^{4}(S, \bbR).
	\end{equation}
	Since $p^{\ast} T^{1,0}Y \cong T^{1,0}S$,
	it is sufficient to prove
	\begin{equation}
	\label{eq:non-triviality-of-second-Chern-class}
		c_{1}(T^{1,0}Y)^{2}
		\notin \Im (H^{2}(Y, \bbR) \xrightarrow{c_{1}(L)} H^{4}(Y, \bbR)).
	\end{equation}
	On the one hand,
	\begin{gather}
		H^{2}(Y, \bbR)
		= \bbR c_{1}(L_{1})
			\oplus \bbR c_{1}(L_{2}), \\
		H^{4}(Y, \bbR)
		= \bbR c_{1}(L_{1})^{2}
			\oplus \bbR c_{1}(L_{1}) c_{1}(L_{2})
			\oplus \bbR c_{1}(L_{2})^{2},
	\end{gather}
	where we use the fact that $n-2 \geq 2$ in the latter equality.
	These imply that
	\begin{equation}
		H^{4}(Y, \bbR)
		= \Im (H^{2}(Y, \bbR) \xrightarrow{c_{1}(L)} H^{4}(Y, \bbR))
			\oplus \bbR c_{1}(L_{2})^{2}.
	\end{equation}
	On the other hand,
	\begin{align}
		c_{1}(T^{1, 0} Y)^{2}
		&= \pqty{ c_{1}(L_{1}) - \frac{n-1}{3} c_{1}(L_{2}) }^{2} \\
		&= \pqty{ c_{1}(L) - \frac{n+2}{3} c_{1}(L_{2}) }^{2} \\
		&= c_{1}(L) \pqty{ c_{1}(L) - \frac{2(n+2)}{3}c_{1}(L_{2}) }
			+ \frac{(n+2)^{2}}{9} c_{1}(L_{2})^{2}.
	\end{align}
	Thus we have \cref{eq:non-triviality-of-second-Chern-class}.
\end{proof}

We next show that \cref{thm:Chern-classes-of-spherical-CR-structure} does not hold in integer coefficients in general.
Note that the following example is the same as that in the proof of~\cite{Takeuchi2020}*{Proposition 4.1}.

\begin{proof}[Proof of \cref{prop:counterexample-to-integer-coefficients}]
	Fix a positive integer $d$.
	Let $h$ be a Hermitian metric on the holomorphic line bundle $\calO(-d)$ over $\cps^{n}$
	such that $\omega \coloneqq - \sqrt{-1} \Theta_{h}$
	is a positive constant multiple of the Fubini-Study form.
	Denote by $(S, T^{1,0}S, \eta)$ the circle bundle associated with $(\cps^{n}, \calO(-d), h)$.
	Since the Fubini-Study metric is Bochner-flat,
	$(S, T^{1,0}S)$ is a spherical CR manifold.
	Let $p \colon S \to Y$ be the canonical projection.
	To simplify notation,
	we write $\tau$ for $c_{1}(\calO(1))$,
	which is a generator of $H^{2}(\cps^{n}, \bbZ) \cong \bbZ$.
	On the one hand,
	consider the Gysin exact sequence
	\begin{equation}
		H^{2}(\cps^{n}, \bbZ) \xrightarrow{-d \cdot \tau}
		H^{4}(\cps^{n}, \bbZ) \xrightarrow{p^{\ast}}
		H^{4}(S, \bbZ) \xrightarrow{}
		H^{3}(\cps^{n}, \bbZ) = 0.
	\end{equation}
	Since $H^{4}(\cps^{n}, \bbZ)$ is freely generated by $\tau^{2}$,
	the above exact sequence implies that $H^{4}(S, \bbZ) \cong \bbZ / d \bbZ$
	and $p^{\ast} \tau^{2}$ is a generator of $H^{4}(S, \bbZ)$.
	On the other hand,
	since $p^{\ast} T^{1,0}\cps^{n}$ is isomorphic to $T^{1,0}S$ as a complex vector bundle,
	\begin{equation}
		c_{1}(T^{1,0}S)
		= (n+1) p^{\ast} \tau,
		\qquad
		c_{2}(T^{1, 0} S)
		= \frac{n(n+1)}{2} p^{\ast} \tau^{2}.
	\end{equation}
	Hence
	\begin{align}
		2(n+2) c_{2}(T^{1,0}S) - (n+1) c_{1}(T^{1,0}S)^{2}
		&= n(n+1)(n+2) p^{\ast} \tau^{2} - (n+1)^{3} p^{\ast} \tau^{2} \\
		&= -(n+1) p^{\ast} \tau^{2}.
	\end{align}
	Therefore if we choose $d$ as a prime integer greater than $n+1$,
	then we have
	\begin{equation}
		 2(n+2) c_{2}(T^{1,0}S)
		 \neq (n+1) c_{1}(T^{1,0}S)^{2}
	\end{equation}
	in $H^{4}(S, \bbZ)$.
\end{proof}

Now we consider the problem whether a given contact manifold admits a spherical CR structure.
We can explicitly construct a Stein fillable contact manifold with no spherical CR structures.

\begin{proposition}
\label{prop:Reinhardt-boundary-without-spherical}
	Let $M$ be the boundary of the strictly pseudoconvex domain
	\begin{equation}
		\Omega
		\coloneqq \Set{ z = (z^{0}, \dots, z^{n}) \in \bbC^{n+1} | \sum_{j = 0}^{n} (\log \abs{z^{i}})^{2} < 1 }
	\end{equation}
	in $\bbC^{n+1}$.
	Then $(M, \Re T^{1,0}M)$ is Stein fillable but admits no spherical CR structures.
\end{proposition}

\begin{proof}
	On the one hand,
	$M$ is diffeomorphic to $S^{n} \times T^{n+1}$.
	Hence its fundamental group $\pi_{1}(M)$ is
	\begin{equation}
		\pi_{1}(M)
		\cong
		\begin{cases}
			\bbZ^{3} & n = 1, \\
			\bbZ^{n+1} & n \geq 2.
		\end{cases}
	\end{equation}
	In particular,
	$\pi_{1}(M)$ is Abelian.
	On the other hand,
	there is a classification of spherical CR manifolds
	with nilpotent fundamental group by Goldman~\cite{Goldman1983}*{Theorem 4.1};
	$M$ must be finitely covered by $S^{2n+1}$, $S^{1} \times S^{2n}$, or $\bbH^{2n+1} / \Gamma$.
	Here $\bbH^{2n+1}$ is the Heisenberg group of dimension $2n+1$
	and $\Gamma \subset \bbH^{2n+1}$ is a discrete cocompact subgroup.
	Considering the universal cover of $M$ yields that
	the first and second cases do not occur.
	Hence $M$ must be finitely covered by $\bbH^{2n+1} / \Gamma$.
	However,
	the fundamental group of $\bbH^{2n+1} / \Gamma$ is $\Gamma$,
	which is non-Abelian;
	see~\cite{Folland2004}*{Section 2} for example.
	This contradicts the fact that $\pi_{1}(M)$ is Abelian.
	Therefore $M$ has no spherical CR structures.
\end{proof}

However,
the total Chern class $c(T^{1,0}M)$ is trivial for the above example.
Our last example is a Stein fillable contact manifold violating the equality \cref{eq:second-Chern-contact-with-spherical}.
Note that the following example is the same as that in the proof of~\cite{Takeuchi2020}*{Proposition 4.2}.

\begin{proof}[Proof of \cref{prop:Stein-fillable-with-no-spherical-CR}]
	Let $\Omega_{0}$ be a Stein domain in a two-dimensional complex manifold $X_{0}$
	such that its boundary $M_{0} \coloneqq \partial \Omega_{0}$
	satisfies $c_{1}(T^{1,0}M_{0}) \neq 0$ in $H^{2}(M_{0}, \mathbb{C})$;
	see~\cite{Etnyre-Ozbagci2008}*{Theorem 6.2} for an example of such $\Omega_{0}$.
	Take a defining function $\rho$ of $\Omega_{0}$
	that is strictly plurisubharmonic near the closure of $\Omega_{0}$.
	Without loss of generality,
	we may assume that $\rho$ is an exhaustion function on $X_{0}$.
	Then,
	for sufficiently small $\epsilon$,
	there exists a diffeomorphism
	$\chi \colon (- \epsilon, \epsilon) \times M_{0}
	\to \rho^{- 1}((- \epsilon, \epsilon))$
	satisfying $\chi(0, p) = p$ and $\rho \circ \chi (t, p) = t$.
	The function $\psi_{0} \coloneqq -1 / \rho$
	gives a strictly plurisubharmonic exhaustion function on $\Omega_{0}$.

	We first show the statement for the case of odd $n$;
	write $n = 2m-1$, $m \geq 2$.
	Consider the domain
	\begin{equation}
		\Omega
		\coloneqq \Set{ (p_{1}, \dots , p_{m}) \in (\Omega_{0})^{m}
		| \psi_{0}(p_{1}) + \dots + \psi_{0}(p_{m}) < 2m / \epsilon }.
	\end{equation}
	The function $\psi(p_{1}, \dots , p_{m}) \coloneqq \psi_{0}(p_{1}) + \dots + \psi_{0}(p_{m})$
	is a strictly plurisubharmonic exhaustion function on $(\Omega_{0})^{m}$,
	and $d \psi \neq 0$ on $M = \partial \Omega$.
	Hence $\Omega$ is a Stein domain
	in $(\Omega_{0})^{m} \subset (X_{0})^{m}$.
	Consider the map $\iota \colon (M_{0})^{m} \to M (\subset (X_{0})^{m})$
	defined by
	\begin{equation}
		\iota(p_{1}, \dots , p_{m})
		\coloneqq (\chi(- \epsilon / 2, p_{1}), \dots , \chi(- \epsilon / 2, p_{m})).
	\end{equation}
	Since this map is homotopic to the natural embedding $(M_{0})^{m} \hookrightarrow (X_{0})^{m}$,
	\begin{equation}
		\iota^{\ast} c(T^{1,0}M)
		= c(\iota^{\ast} T^{1,0}(X_{0})^{m})
		= c(T^{1,0}(X_{0})^{m} |_{(M_{0})^{m}})
		= c((T^{1,0}M_{0})^{m}).
	\end{equation}
	For $1 \leq j \leq m$,
	let $p_{j} \colon (M_{0})^{m} \to M_{0}$ be the $j$-th projection,
	and set
	\begin{equation}
		\tau_{j}
		\coloneqq p_{j}^{\ast} c_{1}(T^{1,0}M_{0}) \in H^{2}((M_{0})^{m}, \bbR).
	\end{equation}
	Then we have
	\begin{equation}
		\iota^{\ast} c(T^{1,0}M)
		= c \pqty{ \bigoplus_{j = 1}^{m} p_{j}^{\ast} T^{1,0}M_{0}}
		= \prod_{j = 1}^{m} p_{j}^{\ast} c(T^{1,0}M_{0})
		= \prod_{j = 1}^{m} (1 + \tau_{j}).
	\end{equation}
	In particular,
	\begin{equation}
		\iota^{\ast} c_{1}(T^{1,0}M)
		= \sum_{j = 1}^{m} \tau_{j},
		\qquad
		\iota^{\ast} c_{2}(T^{1,0}M)
		= \sum_{i < j} \tau_{i} \tau_{j}
	\end{equation}
	From the fact that $c_{1}(T^{1,0}M_{0}) \neq 0$ in $H^{2}(M_{0}, \bbR)$,
	it follows that $\iota^{*} c_{2}(T^{1,0}M) \neq 0$ in $H^{4}((M_{0})^{m}, \bbR)$.
	Moreover,
	by using $\tau_{j}^{2} = 0$,
	we have
	\begin{equation}
		\iota^{\ast} c_{2}(T^{1,0}M)
		= \frac{1}{2} \iota^{\ast} c_{1}(T^{1,0}M)^{2}
		\neq \frac{n+1}{2(n+2)} \iota^{\ast} c_{1}(T^{1,0}M)^{2}.
	\end{equation}
	This implies that
	the contact structure $\xi = \Re T^{1,0}M$ satisfies
	\begin{equation}
		c_{2}(\xi)
		= c_{2}(T^{1,0}M)
		\neq \frac{n+1}{2(n+2)} c_{1}(T^{1,0}M)^{2}
		= \frac{n + 1}{2(n+2)} c_{1}(\xi)^{2}
	\end{equation}
	in $H^{4}(M, \bbR)$.

	We next treat the case of even $n$;
	write $n = 2m$, $m \geq 2$.
	Consider the domain
	\begin{equation}
		\Omega
		\coloneqq \Set{(p_{1}, \dots , p_{m}, z) \in (\Omega_{0})^{m} \times \mathbb{C}
		| \psi_{0}(p_{1}) + \dots + \psi_{0}(p_{m}) + |z|^{2} < 2m / \epsilon}.
	\end{equation}
	This $\Omega$ is a Stein domain in $(\Omega_{0})^{m} \times \mathbb{C} \subset (X_{0})^{m} \times \mathbb{C}$.
	Consider the map $\iota \colon (M_{0})^{m} \to M = \partial \Omega$ given by
	\begin{equation}
		\iota(p_{1}, \dots , p_{m})
		\coloneqq (\chi(- \epsilon / 2, p_{1}), \dots , \chi(- \epsilon / 2, p_{m}), 0).
	\end{equation}
	Then we obtain
	\begin{equation}
		\iota^{\ast} c(T^{1,0}M)
		= c((T^{1,0}M_{0})^{m}).
	\end{equation}
	Similar to the case of odd $n$,
	we have
	\begin{equation}
		c_{2}(\xi)
		\neq \frac{n+1}{2(n+2)} c_{1}(\xi)^{2}
	\end{equation}
	in $H^{4}(M, \bbR)$.
	This proves the statement.
\end{proof}

\section*{Acknowledgements}
The author is grateful to Taiji Marugame and Yoshihiko Matsumoto for helpful comments.

\bibliography{my-reference,my-reference-preprint}

\end{document}


text:
section \<open>\<open>Extra_Vector_Spaces\<close> -- Additional facts about vector spaces\<close>

theory Extra_Vector_Spaces
  imports
    "HOL-Analysis.Inner_Product"
    "HOL-Analysis.Euclidean_Space"
    "HOL-Library.Indicator_Function"
    "HOL-Analysis.Topology_Euclidean_Space"
    "HOL-Analysis.Line_Segment"
    Extra_General
begin

subsection \<open>Euclidean spaces\<close>

typedef 'a euclidean_space = "UNIV :: ('a \<Rightarrow> real) set" ..
setup_lifting type_definition_euclidean_space

instantiation euclidean_space :: (type) real_vector begin
lift_definition plus_euclidean_space ::
  "'a euclidean_space \<Rightarrow> 'a euclidean_space \<Rightarrow> 'a euclidean_space"
  is "\<lambda>f g x. f x + g x" .
lift_definition zero_euclidean_space :: "'a euclidean_space" is "\<lambda>_. 0" .
lift_definition uminus_euclidean_space ::
  "'a euclidean_space \<Rightarrow> 'a euclidean_space"
  is "\<lambda>f x. - f x" .
lift_definition minus_euclidean_space ::
  "'a euclidean_space \<Rightarrow> 'a euclidean_space \<Rightarrow> 'a euclidean_space"
  is "\<lambda>f g x. f x - g x".
lift_definition scaleR_euclidean_space ::
  "real \<Rightarrow> 'a euclidean_space \<Rightarrow> 'a euclidean_space"
  is "\<lambda>c f x. c * f x" .
instance
  apply intro_classes
  by (transfer; auto intro: distrib_left distrib_right)+
end

instantiation euclidean_space :: (finite) real_inner begin
lift_definition inner_euclidean_space :: "'a euclidean_space \<Rightarrow> 'a euclidean_space \<Rightarrow> real"
  is "\<lambda>f g. \<Sum>x\<in>UNIV. f x * g x :: real" .
definition "norm_euclidean_space (x::'a euclidean_space) = sqrt (inner x x)"
definition "dist_euclidean_space (x::'a euclidean_space) y = norm (x-y)"
definition "sgn x = x /\<^sub>R norm x" for x::"'a euclidean_space"
definition "uniformity = (INF e\<in>{0<..}. principal {(x::'a euclidean_space, y). dist x y < e})"
definition "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x'::'a euclidean_space, y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
instance
proof intro_classes
  fix x :: "'a euclidean_space"
    and y :: "'a euclidean_space"
    and z :: "'a euclidean_space"
  show "dist (x::'a euclidean_space) y = norm (x - y)"
    and "sgn (x::'a euclidean_space) = x /\<^sub>R norm x"
    and "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist (x::'a euclidean_space) y < e})"
    and "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. (x'::'a euclidean_space) = x \<longrightarrow> y \<in> U)"
    and "norm x = sqrt (inner x x)" for U
    unfolding dist_euclidean_space_def norm_euclidean_space_def sgn_euclidean_space_def
      uniformity_euclidean_space_def open_euclidean_space_def
    by simp_all

  show "inner x y = inner y x"
    apply transfer
    by (simp add: mult.commute)
  show "inner (x + y) z = inner x z + inner y z"
  proof transfer
    fix x y z::"'a \<Rightarrow> real"
    have "(\<Sum>i\<in>UNIV. (x i + y i) * z i) = (\<Sum>i\<in>UNIV. x i * z i + y i * z i)"
      by (simp add: distrib_left mult.commute)
    thus "(\<Sum>i\<in>UNIV. (x i + y i) * z i) = (\<Sum>j\<in>UNIV. x j * z j) + (\<Sum>k\<in>UNIV. y k * z k)"
      by (subst sum.distrib[symmetric])
  qed

  show "inner (r *\<^sub>R x) y = r * (inner x y)" for r
  proof transfer
    fix r and x y::"'a\<Rightarrow>real"
    have "(\<Sum>i\<in>UNIV. r * x i * y i) = (\<Sum>i\<in>UNIV. r * (x i * y i))"
      by (simp add: mult.assoc)
    thus "(\<Sum>i\<in>UNIV. r * x i * y i) = r * (\<Sum>j\<in>UNIV. x j * y j)"
      by (subst sum_distrib_left)
  qed
  show "0 \<le> inner x x"
    apply transfer
    by (simp add: sum_nonneg)
  show "(inner x x = 0) = (x = 0)"
  proof (transfer, rule)
    fix f :: "'a \<Rightarrow> real"
    assume "(\<Sum>i\<in>UNIV. f i * f i) = 0"
    hence "f x * f x = 0" for x
      apply (rule_tac sum_nonneg_eq_0_iff[THEN iffD1, rule_format, where A=UNIV and x=x])
      by auto
    thus "f = (\<lambda>_. 0)"
      by auto
  qed auto
qed
end

instantiation euclidean_space :: (finite) euclidean_space begin
lift_definition euclidean_space_basis_vector :: "'a \<Rightarrow> 'a euclidean_space" is
  "\<lambda>x. indicator {x}" .
definition "Basis_euclidean_space == (euclidean_space_basis_vector ` UNIV)"
instance
proof intro_classes
  fix u :: "'a euclidean_space"
    and v :: "'a euclidean_space"
  show "(Basis::'a euclidean_space set) \<noteq> {}"
    unfolding Basis_euclidean_space_def by simp
  show "finite (Basis::'a euclidean_space set)"
    unfolding Basis_euclidean_space_def by simp
  show "inner u v = (if u = v then 1 else 0)"
    if "u \<in> Basis" and "v \<in> Basis"
    using that unfolding Basis_euclidean_space_def
    apply transfer apply auto
    by (auto simp: indicator_def)
  show "(\<forall>v\<in>Basis. inner u v = 0) = (u = 0)"
    unfolding Basis_euclidean_space_def
    apply transfer
    by auto
qed
end (* euclidean_space :: (finite) euclidean_space *)

lemma closure_bounded_linear_image_subset_eq:
  assumes f: "bounded_linear f"
  shows "closure (f ` closure S) = closure (f ` S)"
  by (meson closed_closure closure_bounded_linear_image_subset closure_minimal closure_mono closure_subset f image_mono subset_antisym)

lemma not_singleton_real_normed_is_perfect_space[simp]: \<open>class.perfect_space (open :: 'a::{not_singleton,real_normed_vector} set \<Rightarrow> bool)\<close>
  apply standard
  by (metis UNIV_not_singleton clopen closed_singleton empty_not_insert)

end

subset_name: curated


TITLE: Why use ultrasound for medical imaging?
QUESTION [4 upvotes]: What advantage does ultrasound have over sound between 20-20000Hz that it is used in medical imaging over sound in that frequency range?

REPLY [5 votes]: Using waves for imaging, you want to distinguish points in the specimen. Consider the simplest case: you'd like to distinguish 2 discs (or spheres).
The waves are coming from behind the specimen passing through the circle or the waves are reflected on the circle. Considering the phase relationship of the waves at the detector (i.e. investigating interference) because of the diffraction on the 1 circle, even with focusing, one cannot get a single point on the detector. Take a look on the single slit diffraction and on the 2D version, on the Airy disk.
On the focused image you will get concentric circles. For $\alpha$, the angular distance between the two most visible, inner ones one gets
$$\lambda /\left( {2d} \right) = \sin \left( \alpha  \right) \approx \alpha ,$$
where $\lambda$ corresponds to the wavelength and $d$ is the size of disc (therefore, your discs cannot be closer than this). With a huge, $180^\circ$ imaging device one can resolve a distance of $\lambda/2$. It can be seen that the achievable resolution (resolvable smallest distance) is proportional to wavelength.
And as it is known,
$$\lambda  = c/f,$$
where $c$ is the speed of the wave and $f$ is frequency, the higher the frequency, the smaller the wavelength is, increasing the achievable resolution.
Estimation
You can make an estimation on the achievable resolution. You'd like to do measurements in a human body. That consists of water mainly, the speed of sound in the water is $1500m/s$. At $20\text{KHz}$ the resolvable distance is
$$\frac{{1500{\text{m}}/{\text{s}}}}{{2 \cdot 20{\text{kHz}}}} = 3.75{\text{cm}}.$$

We considered only 2D imaging here, but similar applies to 3D where you'd like to get information in 3D. In 3D one measures the time difference between the emission and detection of the signal, and using the $v=s/t$ formula, the distance of the object can be calculated.


TITLE: How do I integrate the integral for the independent Poisson random variable?
QUESTION [0 upvotes]: I don't know how to derive this (my red pen mark)

REPLY [3 votes]: Possible explanation:
Since $\left| -\frac{1}{\sqrt n} \right | \approx 0,$ the interval of integration is very close to zero, and so $x^2 \approx 0 \implies e^{-x^2/2} \approx 1$.


TITLE: identity of a formal power series
QUESTION [0 upvotes]: Recently I encounter an identity $$z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})=z_{1}^{-1}\delta(\frac{z_{0}+z_{2}}{z_{1}})$$  where $\delta(x)=\Sigma_{n\in \mathbb{N}}x^{n}$.
I tried to expand both sides. I get $$ LHS=z_{0}^{-1}\Sigma_{p\in \mathbb{Z}}\Sigma_{q \in \mathbb{N}}(\frac{z_{1}}{z_{0}})^{p-q}(\frac{z_{2}}{z_{0}})^{q}(-1)^{q}$$
$$RHS=z_{1}^{-1}\Sigma_{p\in \mathbb{Z}}\Sigma_{q \in \mathbb{N}}(\frac{z_{0}}{z_{1}})^{p-q}(\frac{z_{2}}{z_{1}})^{q}$$. When I compare the coefficients of both sides, left hand side has negative term. Is this identity wrong or did I miss something?

REPLY [1 votes]: I get $\, \delta(x) = 1/(1-x)\,$ if $\,|x|<1  \,$ and an identity
 $\, z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}}) = \frac1{z_0-z_1+z_2} =
 - z_{1}^{-1}\delta(\frac{z_{0}+z_{2}}{z_{1}}). \,$
For convergence, we need that
 $\, \max(z_1-|z_0|, -|z_1|-z_0) < z_2 < \min(z_1+|z_0|, |z_1|-z_0). \,$


TITLE: Arc lenght of a curve is finite
QUESTION [0 upvotes]: Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$
Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and $\lim_{t \to \infty} \int_0^t |\alpha'(t)|dt$ is finite. Conclude that the arc lenght of the curve is finite.
I've already shown that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$, which implies $\lim_{t \to +\infty} |\alpha'(t)|=(0,0)$.
Since the arclenght of $\alpha$ is defined as $\Gamma(t)=\int_0^t |\alpha'(t)|dt$, then to show it is finite, it is sufficient to show that $\lim_{t \to \infty} \int_0^t |\alpha'(t)|dt$ is finite.
My question is, if a positive function  $f(t) \to 0$ when $t \to +\infty$, is it true that $\int_0^t f(t)dt is finite? If I could show that, then this exercise is a particular case of the statement.
I've directly calculated $\int_0^t |\alpha'(t)|dt=\sqrt{a^2b^2+a^2}\int_o^t e^{bt}dt$ but I don't know if it is useful to know the exact expression of the integral.

REPLY [1 votes]: You have $\alpha'(t) = a e^{bt} (b(\cos t, \sin t) + (-\sin t, \cos t))$,
hence
$|\alpha'(t)| \le a e^{bt}(|b|+1)$.
Since $\int_0^\infty e^{bt} dt = {1 \over |b|}$, we see that $t \mapsto |\alpha'(t)|$ is integrable.
In fact, since you have computed $\int_0^t |\alpha'(x) | dx = a \sqrt{b^2+1} \int_0^t e^{bx} dx $, and $t \mapsto e^{bt}$ is integrable, this is enough to conclude that $\int_0^\infty |\alpha'(x) | dx$ exists. Indeed, it gives the value
$\int_0^\infty |\alpha'(x) | dx = a \sqrt{1 + {1 \over b^2}}$.
Note: It is not sufficient or necessary that $f(t) \to 0$ in order that $f$ be integrable.
For example, $f_1(t) = {1 \over t+1}$ converges to zero, but is not integrable.
Let $\tau(t) = \max(0,1-|t|)$, and consider
$f_2(t) = \sum_{n=1}^\infty \tau(2^n t-2^n)$. Then
$\int_0^\infty f_2(t) dt = \sum_{n=1}^\infty {1 \over 2^n} = 1$, hence $f_2$ is integrable, but $f_2(2^n) = 1$ for all $n$.


TITLE: how to choose the suitable parametric form given a boundary?
QUESTION [0 upvotes]: Find the absolute minimum and maximum values of $g(x, y) = (x^2 + y^2)e^{(−x^2−4y^2)}$ on the set $A = \{(x, y) \in \mathbb R^2 \mid x^2 + 4y^ 2 ≤ 4\}$.
here is the solution //see image

so my question is when you use parametric form how do you know what value to set for $x$ and $y$ given the boundary and why exactly did he choose $x=2\cos\phi$ and $y=\sin\phi$ instead of $x=\cos\phi$ and $y=\sin\phi$ any links to videos will be much appreciated

REPLY [0 votes]: since the boundary is given by the form $x^2+4y^2≤4$ which can be written as $x^2+4y^2=4$
to parametrise an ellipse the following should be applied
given the function for an ellipse $ (x-p)^2/(a)^2 + (y-q)^2/(b)^2 =1$
the parametrisation of the ellipse is as follows:
$ a * cos(t)-p$
$ b * sin(t)-q$
so in this case since $x^2+4y^2=4$ since the right hand side isn't $=1$ we need to divide by whatever its value is to get it to equal 1 hence
$ (x^2)/4+y^2=1$ this is in the form of an ellipse now and can be rewritten as $$(x-0)^2/(2)^2 +(y-0)^2/(1)^2 =1$$ which is equivalent to $$x^2/2^2 +y^2=1$$ and now applying the rule above we get the following $$x=2cos(t)$$ $$y=sin(t)$$
for more information regarding this topic please visit  THIS YOUTUBE LINK


TITLE: Where did I do this min/max problem incorrectly?
QUESTION [1 upvotes]: Current price of a car is \$20,000
The current price changes at a rate of $50-50\sqrt{t}$
When will the price of a new car be at a maximum?
So.... $$\frac{dp}{dt}=p'(t)=50-50\sqrt{t}$$
First, to determine where there is a min/max, I set $f'(x)=0$
(I would then do 2nd deriv. test to determine if its a min or max)
$$50-50\sqrt{t}=0$$
$$50=50\sqrt{t}$$
$$1=\sqrt{t}$$
$$1=t$$
However, t=1 is not a choice.  What have I done wrong?

REPLY [1 votes]: You are doing fine.  For $t \lt 1$ the rate of change is positive, for $t \gt 1$ the rate of change is negative, so $t=1$ is the maximum.


TITLE: Using power series to create a bound
QUESTION [0 upvotes]: I have a question from the following computation of a real integral using a complex integral. The problem writes
$$\frac{1-e^{iz}}{z^2}=\frac{iz}{z^2}+\frac{(iz)^2}{2!z^2}+\frac{(iz)^3}{3!z^3}+\dots$$
Why is it that the remainder term after the $\frac{iz}{z^2}$ is bounded? as $\epsilon \to 0$? This isn't apparent to me.

REPLY [1 votes]: Note that by the power series of the exponential function,
$$
\frac{{1 - e^{iz} }}{{z^2 }} =  - \frac{1}{{z^2 }}\sum\limits_{n = 1}^\infty  {\frac{{(iz)^n }}{{n!}}}  =  - \frac{i}{z} + \sum\limits_{n = 0}^\infty  {i^n \frac{{z^n }}{{(n + 2)!}}}
$$
for all complex $z \neq 0$. Thus if $|z|\leq c$ with some $c>0$, then
$$
\left| {\sum\limits_{n = 0}^\infty  {i^n \frac{{z^n }}{{(n + 2)!}}} } \right| \le \sum\limits_{n = 0}^\infty  {\frac{{\left| z \right|^n }}{{(n + 2)!}}}  \le \sum\limits_{n = 0}^\infty  {\frac{{c^n }}{{(n + 2)!}}}  \le \sum\limits_{n = 0}^\infty  {\frac{{c^n }}{{n!}}}  = e^c .
$$

REPLY [0 votes]: As pointed out
\begin{align*}
E(z)
&=\frac{(iz)^2}{2!z^2}+\frac{(iz)^3}{3!z^2}+\frac{(iz)^4}{4!z^2}+\cdots\\
&=\frac{i^2}{2!}+z\frac{i^3}{3!}+z^2\frac{i^4}{4!}+\cdots\\
&=-\frac{1}{2}+\sum_{k\ge2}z^{k-2}\frac{i^k}{k!}
\end{align*}
the second term of this last expression is a convergent series, whose summands goes to $0$ as $z\to 0$.


TITLE: Choosing $k$ different assignments of binary variables in order to capture the largest volume of the joint probability distribution
QUESTION [1 upvotes]: Assume you have $n$ independent binary variables $\{x_1,\dots,x_n\}$ and for each variable $x_i$ you know that its value is equal to $1$ with a probability $p_i$. I would like to enumerate the joint assignments of these variables in such a way that in $k$ steps I capture as much volume of the joint probability mass function as possible. Does there exist some approximation algorithm which runs faster than the algorithm which enumerates the assignments according to their joint probability $\Pi p_i$. Or something related?

REPLY [0 votes]: You want to find a $k$-subset $S\subseteq\{0,1\}^n$ to maximize
$$\sum_{(y_1,\dots,y_n)\in S} \prod_{i=1}^n \left(p_{i}y_i+(1-p_i)(1-y_i)\right).$$
Equivalently, find the $k$ largest (with multiplicity)
$$\prod_{i=1}^n \left(p_{i}y_i+(1-p_i)(1-y_i)\right).$$
Equivalently, find the $k$ largest (with multiplicity)
$$\sum_{i=1}^n \left((\log p_{i})y_i+(\log(1-p_i))(1-y_i)\right).$$
Some integer linear programming (ILP) solvers have an option to find the $k$ best solutions.  An alternative approach to get the $k$ best solutions is to call an ILP solver $k$ times.  After each solve, yielding optimal solution $\hat{y}\in\{0,1\}^n$, impose an additional no-good cut
$$\sum_{i:\hat{y}_i = 0} y_i + \sum_{i:\hat{y}_i = 1} (1 - y_i) \ge 1$$
that excludes only that solution.


TITLE: Find the image of an element without clear function rule.
QUESTION [2 upvotes]: How can I find $f(4)$ from the following equation:
$$f(x) + f(2x+1) = 6x+10$$

REPLY [12 votes]: Let $g(x)=f(x)-2x-4$ to get
$$g(x)+g(2x+1)=0$$
Substitute $x\mapsto2^x-1$ and let $h(x)=g(2^x-1)$ to get
$$h(x)+h(x+1)=0$$
We can produce infinitely many solutions for $h$. An obvious form can be given by trigonometric functions, and more generally Fourier series. As a simple example:
$$h(x)=A\sin(\pi x)$$
$$g(x)=A\sin(\pi\log_2(x+1))$$
$$f(x)=2x+4+A\sin(\pi\log_2(x+1))$$
$$f(4)=12+A\sin(\pi\log_2(5))$$
which depends on the arbitrary value $A$. Thus there is no unique solution, given the conditions.


TITLE: Why does a teeter-totter (see-saw) tend to incline towards the heavier end?
QUESTION [3 upvotes]: Why does a teeter-totter (see-saw) tend to incline towards the heavier end? Since objects of different mass tend to fall at the same speed (assuming a vacuum), why do then heavier objects push harder on a scale? Why are they heavier? Even though they have different masses, their "resistance" against Earth's pull is proportional. Right?
This seems like a very basic question.

REPLY [2 votes]: The reason that objects tend to fall at the same rate is because on earth we can approximate the force of gravity to be $F=mg$.  In this case, the force is really $$|F|=G\frac{Mm}{r^2}$$ where $r$ is the distance to the center of the earth, $M$ is the mass of the earth, and $G$ is the gravitational constant.  Since these things tend to be the same near the earths surface, we approximate $g = \frac{GM}{r^2}$.  This we can relate to $F=ma$ (Newton's Laws), and we can infer that for objects of any mass, the acceleration is the same, at $a=g$.
Now, to the point about the see-saw.  The accelerations are the same, but the forces are not the same, since they are also proportional to the mass, $m$.  Where a see-saw balances at is dependent on the torque, which is given by $\tau = \vec{r}\times\vec{F}$, where $\vec{r}$ is the distance from the pivot point.  Now, for masses the same distance away from the pivot point, the torque is greater for the greater force.  For objects under the influence of gravity, the greater force belongs to the object with greater mass.  Therefore, the see-saw tends to tip toward the side with greater mass.

REPLY [0 votes]: When you say "push harder" you are talking about force, which makes objects move. The manner in which they move is governed by Newton's second law,
$$F_{net}=ma,$$
where $m$ is the mass of the object and $a$ is the acceleration. Now, in a gravitational field, the weight of an object is equal to it's mass times the gravitational field of the earth, $g$. So the net force is simply $mg$. But if we put that into Newton's second law,
$$F_{net}=mg=ma\rightarrow a=g$$
So, the acceleration is constant, exactly as you expect.
The see-saw works a bit differently. A see-saw is fixed to a point in the center (I'll call that the pivot point), so it will never move (unless you put something way too heavy, and then it will break!). However, the see-saw can still rotate, and rotations are caused by torques. The torque $\tau$ due to a force $F$ acting a distance $r$ from the point of rotation is
$$\tau=rF$$
(I am assuming this see-saw is perfectly horizontal, see details below). The further the force is from the pivot, the greater the torque.
In this case, our force is caused by gravity, so each mass is applying a force $mg$ to each end of the see-saw. If you have two masses on either side of a see-saw, they will be balanced if the two torques are equal to each other,
$$r_1m_1g=r_2m_2g$$
So if one mass is smaller $m_1<m_2$, it must be placed a farther distance away $r_1>r_2$ in order for the system to remain balanced.
Details: Actually, torque is a bit more complicated. It depends on the angle $\phi$ between the application of the force and the vector from the pivot to the place the force is acting:
$$\tau=rF\sin\phi.$$
That's why I made sure the see-saw was horizontal, because the force of gravity will be straight downwards and I don't have to worry about that angle.
EDIT: More details, so everyone understands that these answers are all the same. The actual definition of torque is indeed a cross-product, $\vec{\tau}=\vec{r}\times\vec{F}$, I'm just talking about the magnitude $\tau=rF\sin\phi$.

REPLY [0 votes]: I'm not sure exactly what you are asking, but this may help: Let's say gravity causes a blob of clay to push downward on a weight scale with a force of 1 Newton. Now let's say we get another blob of clay identical to the first and add it to the scale along with the first blob of clay, so there are two blobs of clay on the scale. The downward force will be 2 Newtons because force is additive--the total force is the sum of the forces from each blob (as long as the force is in the same direction, of course). Now smoosh those two blobs of clay together so that its now one big blob and keep the big blob on the scale. You haven't changed anything about the two blobs of clay except they are stuck together. So the two blobs, now in the form of one blob, still push with 2 Newtons. Also, consider that two identical blobs released at the same time will fall at the same rate that a single blob, by itself, will fall. Now connect the blobs together with a light string and again drop them at the same time. Surely that string won't cause them to suddenly fall faster, so totally smooshing them together should also not change how fast they fall, because you can still think of them as two blobs that just happen to be connected together.


TITLE: Divisibility exercises
QUESTION [1 upvotes]: Let $p>2$ be a prime number and $n\in\{1,2,\dots, p-2\}$. I have to prove that $p$ does not divide $2^{n}-1$. Any idea? I have tried using the Fermat's theorem, which says that $p$ divides $2^{p}-1$, but I don't get anywhere...

REPLY [1 votes]: If $p=7$ the allowable $n$ are 1,2,3,4,5 but $7$ divides $2^3-1.$

REPLY [0 votes]: The result does not hold. Let $p=7$ and $n=3$. There are infinitely many other examples.
Added: If $p$ is a prime $\gt 3$ for which $2$ is not a primitive root, then there is an $n$ between $1$ and $p-2$ such that $p$ divides $2^n-1$. In particular, this is the case whenever $2$ is a quadratic residue of $p$. Thus every prime of the form $8k\pm 1$ provides a counterexample. There are others.


TITLE: cover and refinement (Godement Chapter 3 Q4)
QUESTION [0 upvotes]: Godement's definition of a cover of a set $X$ is a family $(U_i)_{i\in I}$ of subsets of $X$ whose union is $X$ (as opposed to the usual inclusion).
Keeping that in mind, here is the question:

Given two covers of a set $X$, show there exists a refinement of each one.

For two covers $(U_i)_{i\in I}$ and $(V_j)_{i\in J}$ of $X$, my idea is to take the union of all possible intersections between the elements of the two families (i.e. intersect all $U_i$'s with all the $V_j$'s then take the union) and check if it is a cover of $X$. I am writing this as $\bigcup_\limits{i\in I}\bigcup\limits_{j\in J}\big(U_i\bigcap U_j\big)$.
So I have two questions:
1) Is what I have written indeed the union of all intersections between elements of the two covers?
2) Is the the union of all intersections a cover of $X$? + A hint on how to show why / why not.

REPLY [0 votes]: 1) The notation is fine, a condensed one: $\bigcup\limits_{i\in I, j\in J}\big(U_i\bigcap V_j\big)$
2) Let $x\in \bigcup_{j\in J}\big(U_1\bigcap V_j\big)$, this is equivalent to saying there exists $j\in J$ such that $x\in \big(U_1\bigcap V_j\big)$, i.e. $x\in U_1$ and there is $j\in J$ such that $x\in V_j$ which is the same as saying $x\in U_1$ and $x\in \bigcup_{j\in J}V_j$. Now this is equivalent to $x\in U_1\bigcap \big(\bigcup_{j\in J}V_j\big)$. Do the same for each $U_i$ and we get $\bigcup\limits_{i\in I, j\in J}\big(U_i\bigcap V_j\big) = \big(\bigcup_{i\in I}U_i\big)\bigcap \big(\bigcup_{j\in J}V_j\big)$. And since each family is a cover of $X$, this intersection is also cover of $X$.


TITLE: What energy conversion takes place with gravitational waves?
QUESTION [2 upvotes]: A black hole binary consists of two black holes orbiting their barycenter. Gravitational waves make the pair lose the kinetic energy. The black holes eventually spiral into each other and merge into a larger black hole.
I don't have the level of understanding the exact process behind that. I know that celestial bodies may lose kinetic energy for reasons like tidal forces, collisions and gravitational perturbation of other bodies. Tidal forces may cause friction at the planet's crust and convert kinetic energy into heat. That is why, for instance the moon is tidally locked to Earth because tidal forces made it lose rotation energy that turned into heat.
The question is: Gravitational waves make the two black holes lose kinetic energy, but this energy is converted into what? If there is no other celestial body nearby that will disturb them,  then where this energy goes to? Is it transferred to the space-time fabric?

REPLY [3 votes]: The question is: Gravitational waves make the two black holes lose kinetic energy, but this energy is converted into what? If there is no other celestial body nearby that will disturb them, then where this energy goes to? Is it transferred to the space-time fabric?

Classical gravitational waves are a distortion on the space time, and if they cross matter they lose part of their energy in vibrations/kinetic-energy as we have seen in the LIGO detectors, which detected the waves by the tiny motions of the detectors.
If no matter, cosmic dust to stars and galaxies is met on their way they will go off to infinity.
Alternatively, effective quantization of gravity tells us that the classical gravitational waves are built up by gravitons, the analogue of photons in electromagnetism. Similar to photons, some will lose energy by interacting gravitationally with matter, and if they meet nothing on the way they will move off to infinity.


TITLE: Constructing an explicit tower with quadratic extensions.
QUESTION [1 upvotes]: I am so confused as to where to even start with this problem:
a, b, c, and d are real numbers.
Let α = a + bi, β = c + di, such that α = β2.
Let Q be the set of rational numbers.  Let K be a field, such that Q ⊆ K and a, b ∈ K:
K0 ⊂ K1 ⊂ K2 ⊂ ... ⊂ Kr
K = K0
My job is to find r and construct an explicit tower while considering the following:
All [Kj+1 : Kj ] = 2
c, d ∈ Kr
All Ki contain real elements (thus, we can't extend by i).
I've been trying to work with norms, but I just don't see it right now.  I'm hoping to implement the following theorem:
If F and K are both fields, with [F:K] = 2, and characteristic of K not equal to 2, then ∃α ∈ F such that F = K(α) and α2 = β ∈ K
A nudge in the right direction would be much appreciated.  Thank you for your time!

REPLY [1 votes]: Consider the sequence of primes $(p_n)_{n \geq 1}$. Then the following is what you want,
$$\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},...) = \bigcup_{n=1}^{\infty} \mathbb{Q}(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_n})$$


TITLE: Measuring equal distances between points on a line
QUESTION [0 upvotes]: I have a 36 inch piece of wood (Length) I will placing 6 coat hanger hooks along the length of the wood.  The first hook will be placed 1 inch down the length from one end and the last hook will be placed 1 inch from the other end of this 36 inch piece of wood.  Question how to find the equal distance between the other 4 hooks to be placed on this wood. All six hooks need to be spaced evenly.

REPLY [1 votes]: Here's a simple formula for figuring this out.
(A-(Bx2))/(C-1)
where:
A = Length of your board
B = Distance of First Hook From Edge of Board
C = Total Number of Hooks
Here is an Excel 2007 spreadsheet that will do the work for you. All you have to do is enter the values of A, B, & C.
Hope this helps!


TITLE: Limit of $u_{n}=\frac{1}{n !} \sum_{k=0}^{n} k !$
QUESTION [5 upvotes]: I encountered a sequence $(u_n)_{n \in \mathbb{N}} $ defined as
$$ u_{n}=\frac{1}{n !} \sum_{k=0}^{n} k ! $$
And I wonder what is the limit. It seems to be 1 but even Wolfram Alpha cannot figure it out.
My first idea was to write $$ \frac{k!}{n!} = \frac{1}{(k+1)...(n-1)n}$$ and
$$  (k+1)^{n-k} \leq (k+1)...(n-1)n \leq n^{n-k} $$
EDIT : I found a solution but Yanko found something more elementary.
My solution :
\begin{align}
  u_n &= 1 + \frac{1}{n} + \frac{1}{n(n-1)} + \frac{1}{n(n-1)(n-2)} + ... + \frac{1}{n!}   \\
&= 1 + \frac{1}{n} + (n-1) \times o(\frac{1}{n^2})  \\
&= 1 + \frac{1}{n}  + o(\frac{1}{n}) \\
\end{align}
Thus all terms converge towards 0 except the first one, and the limit is 1.

REPLY [1 votes]: By the Stolz–Cesàro theorem,
$$
\lim_{n\to\infty} u_n = \lim_{n\to\infty}\frac{n!}{n! - (n-1)!} = \lim_{n\to\infty}\frac{1}{1-1/n} = 1.
$$


text:
\begin{document}

\begin{abstract}
A partial difference set $S$ in a finite group $G$ satisfying $1 \notin S$ and $S = S^{-1}$ corresponds to an undirected strongly regular Cayley graph ${\rm Cay}(G,S)$.  While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian.  In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center.  In particular, these results apply to $p$-groups, and we are able to rule out the existence of partial difference sets in many instances.
\end{abstract}

\maketitle

\section{Introduction}
\label{sect:intro}

A subset $S$ of a finite group $G$ is a $(v, k, \lambda, \mu)$-\textit{partial difference set} (PDS) of $G$ if $|G| = v$, $|S| = k$, and each nonidentity element $g \in G$ can be written in either $\lambda$ or $\mu$ different ways (depending on whether or not $g$ is in S) as $g = ab^{-1}$, where $a,b \in S$, and, following \cite{M}, $S$ is said to be a \textit{regular PDS} if $1 \notin S$ and $S^{-1} = S$ (i.e., $s^{-1} \in S$ whenever $s \in S$).

Part of the motivation to find examples of regular partial difference sets is the connection with strongly regular Cayley graphs.  Given a subset $S$ of the group $G$, the \textit{Cayley graph} $\Cay(G,S)$ is defined to be the graph with vertex set the elements of $G$ such that $g, h \in G$ are adjacent if and only if $gh^{-1} \in S$.  If the set $S$ is a $(v, k, \lambda, \mu)$-PDS, then the Cayley graph $\Cay(G,S)$ is a $(v, k, \lambda, \mu)$-\textit{strongly regular graph} (SRG) \cite[Proposition 1.1]{M}, which means that $\Cay(G,S)$ has $v$ vertices, $\Cay(G,S)$ is regular of degree $k$, any two adjacent vertices in $\Cay(G,S)$ have exactly $\lambda$ common neighbors, and any two nonadjacent vertices in $\Cay(G,S)$ have exactly $\mu$ common neighbors.  For further applications of partial difference sets to coding theory and finite geometry, see the survey of Ma \cite{M}.

The case when $G$ is abelian has been thoroughly studied; see \cite{M} for a survey of older results and \cite{DKW, DNW, DW1, DW2, FMX, MS, MX, Polhill, W1, W2} for a number of very recent results.  On the other hand, comparatively little is known in the case when $G$ is nonabelian.  There have been constructions in sporadic cases (see, for instance, \cite{GMV, JK}) and some instances of constructions of infinite families (see \cite{FHC, FL,  Swartz}).  At the same time, there have been relatively few results dealing with the nonabelian case in general.  For instance, in the abelian case, there are restrictions such on \textit{(numerical) multipliers} of the set $S$ (see \cite[Theorem 4.1]{M}), whereas in the general case when the group is allowed to be nonabelian, the corresponding restriction is weaker (see \cite[Theorem 4.3]{M}, originally published in \cite{GL}).

The purpose of this paper is to provide new restrictions on partial difference sets that apply to nonabelian groups as well as abelian groups.  In \cite{Yoshiara}, Yoshiara was able to provide restrictions on groups acting regularly on the point set of a finite \textit{generalized quadrangle} (see Subsection \ref{sub:pggq}).  Since the \textit{collinearity graph} of a generalized quadrangle is a strongly regular graph, this provides restrictions on the strongly regular Cayley graphs (and, hence, partial difference sets) in this specific instance.  One of the main ingredients in Yoshiara's analysis was Benson's Lemma (see Lemma \ref{lem:Benson}), which has since been generalized to strongly regular graphs by DeWinter, Kamischke, and Wang \cite{DKW}.  The main idea of this paper is to generalize the results of Yoshiara to general strongly regular graphs using this recent result of DeWinter, Kamischke, and Wang.

This paper is organized as follows.  In Section \ref{sect:prelim}, we introduce the preliminary combinatorial results about strongly regular graphs and summarize Yoshiara's results from \cite{Yoshiara}.  In Section \ref{sect:YoshSRG}, we generalize Yoshiara's results to strongly regular graphs.  We apply these results to \textit{partial geometries} in Section \ref{sect:pg}, and we produce simple numerical conditions in Section \ref{sect:rest} that rule out the existence of partial difference sets in many instances, especially when the group has a nontrivial center.  Finally, in Section \ref{sect:infeas} we include a table of parameters of partial difference sets that are ruled out in groups having a nontrivial center using the results in Section \ref{sect:rest}.

\section{Preliminaries}
\label{sect:prelim}

\subsection{Strongly regular graphs and their parameters}

Let $\Gamma$ be a $(v,k,\lambda, \mu)$-strongly regular graph.  If $\Gamma$ has adjacency matrix $A$, then $A$ has eigenvalues $\nu_1 = k$,
\[\nu_2 = \frac{1}{2}\left(\lambda - \mu + \sqrt{\Lambda} \right), \quad \nu_3 = \frac{1}{2}\left(\lambda - \mu - \sqrt{\Lambda} \right),\]
where $\Lambda = (\lambda - \mu)^2 + 4(k - \mu) = (\nu_2 - \nu_3)^2$.  A \textit{conference graph} is a strongly regular graph with $k = (v-1)/2$, $\lambda = (v-5)/4$, and $\mu = (v-1)/4$.  If the strongly regular graph $\Gamma$ is not a conference graph, then $2k + (v-1)(\lambda - \mu) \neq 0$, and the eigenvalues $\nu_2$ and $\nu_3$ are integers.

The following lemma collects some well-known results about the relationships among the parameters of a strongly regular graph.  (See, for instance, \cite{BCN} or \cite{GodsilRoyle}.)  These results are used freely later in the paper.

\begin{lem}
 \label{lem:SRGparameters}
 Let $\Gamma$ be a $(v,k,\lambda, \mu)$-strongly regular graph with eigenvalues $k, \nu_2, \nu_3$, where $\nu_2 > \nu_3$.  Then, the following hold:
 \begin{itemize}
  \item[(1)] $k(k - \lambda - 1) = (v - k - 1)\mu$;
  \item[(2)] $k = \mu - \nu_2 \nu_3$;
  \item[(3)] $\mu v = (k - \nu_2)(k - \nu_3)$, and, if $\mu > 0$, then $v = (k - \nu_2)(k - \nu_3)/\mu$;
  \item[(4)] $\lambda = \mu + \nu_2 + \nu_3$;
  \item[(5)] the complement of $\Gamma$ is a $(v, v-k-1, v -2k + \mu -2, v - 2k + \lambda)$-strongly regular graph with eigenvalues $-\nu_3 - 1$ and $-\nu_2 - 1$.
 \end{itemize}
\end{lem}

\subsection{Partial geometries and generalized quadrangles}
\label{sub:pggq}

A \textit{partial geometry} $\pg(s,t, \alpha)$ is an incidence geometry of points and lines satisfying the following three conditions:

\begin{itemize}
 \item[(i)] each point is incident with $t + 1$ lines, and two points are mutually incident with at most one line;
 \item[(ii)] each line is incident with $s + 1$ points, and two lines are mutually incident with at most one point;
 \item[(iii)] if $P$ is a point and $\ell$ is a line not incident with $P$, there are exactly $\alpha$ points on $\ell$ collinear with $P$.
\end{itemize}

If $\calS$ is a $\pg(s,t, \alpha)$, then we say it has order $(s,t, \alpha)$.  If $\alpha = 1$, then the partial geometry is a \textit{generalized quadrangle} of order $(s,t)$.

Given a partial geometry $\calS$ of order $(s,t, \alpha)$, one may define the associated \textit{collinearity graph} $\Gamma_{\calS}$, which has vertex set the points of $\calS$ and edge set consisting of pairs of collinear points.  It is well known that the collinearity graph of a $\pg(s,t,\alpha)$ is an $((s+1)(st + \alpha)/\alpha, s(t+1), s - 1 + t(\alpha - 1), \alpha(t+1))$-strongly regular graph.

\subsection{Automorphisms of generalized quadrangles and strongly regular graphs}

In this subsection, we present some of the main ideas of Yoshiara in \cite{Yoshiara} and the recent result of DeWinter, Kamischke, and Wang \cite{DKW}, which is a generalization of Benson's Lemma to arbitrary strongly regular graphs.  Throughout this paper, if $G$ is a group and $x \in G$, then $x^G$ denotes the conjugacy class of $G$ containing $x$ and $C_G(x)$ denotes the centralizer of $x$ in $G$.

\begin{lem}\cite[Lemma 3]{Yoshiara}
\label{lem:Yoshiara1}
 Suppose the group $G$ acts regularly on the point set $\calP$ of a generalized quadrangle $\calQ$ of order $(s,t)$.  Fix a distinguished point $O$ of $\calQ$.  Let $a \in G$ be a nontrivial automorphism, $\Delta := \{g \in G: O^g \sim O\} \cup \{1\}$, $\Delta^c$ be the set-theoretic complement of $\Delta$ in $G$, and let $d_1(a)$ be the number of points that mapped to collinear points but are not fixed by $a$.  Then,
 \[d_1(a) = |a^G \cap \Delta| |C_G(a)| = (s+1)(t+1) + (s+t)u_a \]
 for some integer $u_a$,  Furthermore,
 \[ |a^G \cap \Delta^c| |C_G(a)|= t(s-1)(s+1) - (s+t)u_a.\]
\end{lem}

\begin{lem}\cite[Lemma 6]{Yoshiara}
\label{lem:Yoshiara2}
 Suppose the group $G$ acts regularly on the point set $\calP$ of a generalized quadrangle $\calQ$ of order $(s,t)$, let $r = \gcd(s,t)$, and let $a$ be a nontrivial element of $G$.  For a distinguished point $O$ of $\calQ$, define $\Delta := \{ g \in G: O^g \sim O\} \cup \{1\}$.  Then, the following hold.
 \begin{itemize}
  \item[(1)] If $r > 1$, then $a^G \cap \Delta \neq \varnothing$.
  \item[(2)] $|a^G \cap \Delta^c|$ is a multiple of $r$ (possibly equal to $0$).
 \end{itemize}
\end{lem}

Lemma \ref{lem:Yoshiara2} follows almost immediately from Lemma \ref{lem:Yoshiara1}.  One of the main ingredients in the proof of Lemma \ref{lem:Yoshiara1} is Benson's Lemma.

\begin{lem}\cite[Lemma 4.3]{Benson}
\label{lem:Benson}
 If $x$ is an automorphism of a finite generalized quadrangle of order $(s,t)$, $d_0(x)$ denotes the number of fixed points of $x$, and $d_1(x)$ denotes the number of points that are sent to collinear points by $x$ (but are not fixed), then
 \[(t+1)d_0(x) + d_1(x) \equiv (st+1) \pmod {s+t}.\]
\end{lem}

Benson's Lemma is a specific instance of a technique, attributed to Graham Higman, which calculates the value of a character of the automorphism group of an association scheme on an eigenspace; see \cite[pp. 89--91]{CameronPerm}.  In particular, a generalization of Benson's Lemma has been proved by De Winter, Kamischke, and Wang for non-conference strongly regular graphs.

\begin{lem}\cite[Theorem 1]{DKW}
\label{lem:DKW}
Let $\Gamma$ be a $(v,k, \lambda, \mu)$-strongly regular graph whose adjacency matrix has integer eigenvalues $k, \nu_2$, and $\nu_3$, where $\nu_2 > \nu_3$.  If $x$ is a nontrivial automorphism of $\Gamma$ that such that $x$ fixes $d_0(x)$ vertices of $\Gamma$ and sends $d_1(x)$ vertices to adjacent vertices, then
\[k - \nu_3 \equiv -\nu_3 d_0(x)  + d_1(x) \pmod {(\nu_2 - \nu_3)}. \]
\end{lem}

\section{Generalizing Yoshiara's results to strongly regular graphs}
\label{sect:YoshSRG}

Throughout this section, we will assume that $\Gamma$ is a $(v,k,\lambda, \mu)$-strongly regular graph with eigenvalues $k$, $\nu_2$, and $\nu_3$, with $\nu_2 > \nu_3$.  We will assume that the group $G$ acts regularly on the vertex set of $\Gamma$, and, for a fixed vertex $\alpha$ of $\Gamma$, we let $\Delta = \{g \in G: \alpha^g \sim \alpha\} \cup \{1\}$.  For a nontrivial element $x \in G$, we define $d_1(x)$ to be the number of vertices sent to adjacent vertices by $x$.

By replacing Benson's Lemma (Lemma \ref{lem:Benson}) with the generalization by De Winter, Kamischke, and Wang (Lemma \ref{lem:DKW}), we are able to prove a generalization of Lemma \ref{lem:Yoshiara1} for non-conference strongly regular graphs.

\begin{lem}
 \label{lem:main1}
If $x$ is a nontrivial element of $G$, then
\[d_1(x) = k - \nu_3 + u_x(\nu_2 - \nu_3) = \mu - \nu_3(\nu_2 + 1) + u_x(\nu_2 - \nu_3) = |x^G \cap \Delta| |C_G(x)|\]
for some integer $u_x$.  Furthermore, assuming $\mu > 0$, we have
\[|x^G \cap \Delta^c||C_G(x)| = \frac{\nu_2\nu_3(\nu_2 + 1)(\nu_3 + 1)}{\mu} - \nu_2(\nu_3 + 1) - u_x(\nu_2 - \nu_3).\]
\end{lem}

\begin{proof}
 Let $\Gamma$ be such a $(v,k,\lambda, \mu)$-strongly regular graph with regular group of automorphisms $G$, and let $x$ be a nontrivial element of $G$.  Since $x$ does not fix any vertices, it follows by Lemma \ref{lem:DKW} that, for some integer $u_x$,
 \[ d_1(x) = k - \nu_3 + u_x(\nu_2 - \nu_3).\]  Equivalently, using Lemma \ref{lem:SRGparameters} (2), we see that
 \[ d_1(x) = \mu - \nu_3(\nu_2 + 1) + u_x(\nu_2 - \nu_3).\]
 Now, since $G$ acts regularly on the vertices of $\Gamma$, for a fixed vertex $\alpha$, we have $V(\Gamma) = \{ \alpha^g : g \in G\}.$  Suppose $\alpha^g$ is sent to an adjacent vertex by $x$, i.e., suppose $\alpha^{gx} \sim \alpha^g$.  This happens if and only if $\alpha^{gxg^{-1}} \sim \alpha$, which implies that $gxg^{-1} \in x^G \cap \Delta$.  Since $gxg^{-1} = hxh^{-1}$, where $g,h \in G$, if and only if $g^{-1}h \in C_G(x)$, we see that $d_1(x) = |x^G \cap \Delta||C_G(x)|$.  Finally, since $|x^G \cap \Delta^c| = |x^G| - |x^G \cap \Delta|$ and
 \[|x^G||C_G(x)| = v = \frac{(k - \nu_2)(k-\nu_3)}{\mu} = \frac{(\mu - \nu_2 \nu_3 - \nu_3)(\mu - \nu_2\nu_3 - \nu_3)}{\mu}, \]
 we have
 \[ |x^G \cap \Delta^c||C_G(x)| = \frac{\nu_2\nu_3(\nu_2 + 1)(\nu_3 + 1)}{\mu} - \nu_2(\nu_3 + 1) - u_x(\nu_2 - \nu_3),\]
 as desired.
\end{proof}

\begin{lem}
 \label{lem:main2}
 If $(\nu_2 - \nu_3)$ does not divide $\mu - \nu_3(\nu_2 + 1)$ and $x$ is a nontrivial element of $G$, then $x^G \cap \Delta \neq \varnothing$.
\end{lem}

\begin{proof}
 If $x^G \cap \Delta = \varnothing$, then, by Lemma \ref{lem:main1},
 \[\mu - \nu_3(\nu_2 + 1) + u_x(\nu_2 - \nu_3) = 0. \]
 The result follows.
\end{proof}

\begin{lem}
 \label{lem:main3}
 Let $x$ be any nontrivial element of $G$, let $r = \gcd(\nu_2(\nu_3 + 1), \nu_3(\nu_2 + 1))$, and assume $\gcd(r, \mu) = 1$.  Then, the following hold.
 \begin{itemize}
  \item[(1)] If $r > 1$, then $x^G \cap \Delta \neq \varnothing$.
  \item[(2)] $|x^G \cap \Delta^c|$ is a multiple of $r$ (possibly equal to $0$).
 \end{itemize}
\end{lem}

\begin{proof}
 We will prove (1) first.  If $x^G \cap \Delta = \varnothing$, then, by Lemma \ref{lem:main1}, it follows that
 \[ 0 = |x^G \cap \Delta| |C_G(x)| = \mu - \nu_3(\nu_2 + 1) + u_x(\nu_2 - \nu_3)\]
 for some integer $u_x$.  Noting that $\nu_2 - \nu_3 = \nu_2(\nu_3 + 1) - \nu_3(\nu_2 + 1)$, we have $r$ divides $\mu$.  However, by assumption, $\mu$ is coprime to $r$ and $r > 1$, a contradiction.  Hence, $x^G \cap \Delta \neq \varnothing$.

 To prove (2), we note first that, since $\gcd(r, \mu) = 1$, $r$ divides $\nu_2 \nu_3 (\nu_2 + 1)(\nu_3 + 1)/\mu$.  Since $r$ divides both $\nu_2(\nu_3 + 1)$ and $\nu_2 - \nu_3$, it follows from Lemma \ref{lem:main1} that $|x^G \cap \Delta^c| |C_G(x)|$ is a multiple of $r$.  On the other hand,
 \[|G| = v = \mu - \nu_2(\nu_3 + 1) - \nu_3(\nu_2 + 1) + \frac{\nu_2 \nu_3 (\nu_2 + 1)(\nu_3 + 1)}{\mu}. \]
 Since $r$ is coprime to $\mu$ but divides each other term, $r$ is coprime to $|G|$ and hence also to $|C_G(x)|$, implying that $r$ divides $|x^G \cap \Delta^c|$, as desired.
\end{proof}

\section{An application to partial geometries}
\label{sect:pg}

Groups acting on partial geometries have been studied before, see \cite{DeWinter, LMS, TTV}.  In particular, an analogue of Benson's Lemma was proved for partial geometries in \cite{DeWinter}, and the following result was obtained for abelian groups acting regularly on the point set of a partial geometry.

\begin{lem}\cite[Corollary 2.3]{DeWinter}
 If $\calS$ is a partial geometry $\pg(s,t, \alpha)$, $\alpha \neq s+1$, and $\calS$ admits an abelian point-regular group of automorphisms, then
 \[(s+1)\frac{st+\alpha}{\alpha} \equiv (s+1)(t+1) \equiv 0 \pmod {s+t - \alpha + 1}. \]
\end{lem}

In \cite{TTV}, the following result is proved.

\begin{lem}\cite[Corollary 4]{TTV}
 Let $\calS$ be a partial geometry of order $(s, t, \alpha)$, and let $x$ be an automorphism of $\calS$.  If $s$, $t$, and $\alpha - 1$ have a common divisor distinct from $1$, then there exists at least one fixed point or at least one point which is mapped to a point collinear to itself.
\end{lem}

We are able to apply Lemmas \ref{lem:main1}, \ref{lem:main2}, and \ref{lem:main3} to obtain similar results.  Note that $d_1$ and $\Delta$ are as defined in Section \ref{sect:YoshSRG}.

\begin{prop}
 \label{prop:pg}
 Let $G$ act regularly on the point set of a partial geometry $\calS$ of order $(s,t,\alpha)$.  If $x$ is a nontrivial element of $G$, then the following hold.

\begin{itemize}
 \item[(1)] \[d_1(x) = (s+1)(t+1) + u_x(s + t - \alpha + 1) = |x^G \cap \Delta||C_G(x)|,\]
 where $u_x$ is an integer, and, furthermore,
 \[|x^G \cap \Delta^c||C_G(x)| = \frac{t(s+1)(s - \alpha)}{\alpha} - u_x(s + t - \alpha + 1). \]
 \item[(2)] If $s + t - \alpha + 1$ does not divide $(s+1)(t+1)$, then $x^G \cap \Delta \neq \varnothing$.
 \item[(3)] Let $r = \gcd(t(s - \alpha), s + t - \alpha + 1)$, and suppose $r > 1$ and coprime with $\alpha(t+1)$.  Then $x^G \cap \Delta \neq \varnothing$ and $|x^G \cap \Delta^c|$ is a multiple of $r$ (possibly equal to $0$).
\end{itemize}
\end{prop}

\begin{proof}
 Noting that the collinearity graph of $\calS$ is an $((s+1)(st + \alpha)/\alpha, s(t+1), s - 1 + t(\alpha - 1), \alpha(t+1))$-strongly regular graph, the results for (1), (2), and (3) follow immediately from Lemmas \ref{lem:main1}, \ref{lem:main2}, and \ref{lem:main3}, respectively.
\end{proof}

\section{New restrictions on parameters of partial difference sets}
\label{sect:rest}

While partial geometries and generalized quadrangles do not have natural complements, strongly regular graphs do.  Moreover, since a group of automorphisms preserves both edges and non-edges, a group acting regularly on a strongly regular graph will also act regularly on its complement.  Immediately, this yields the following result.  We keep the notation from Section \ref{sect:YoshSRG} for the following results.

\begin{prop}
 \label{prop:comp}
 Let $x$ be a nontrivial element of $G$, which acts regularly on the non-conference $(v,k, \lambda, \mu)$-strongly regular graph $\Gamma$.  If $\nu_2 - \nu_3$ does not divide either of $\mu - \nu_3(\nu_2 + 1)$ or $v - 2k + \lambda - \nu_3(\nu_2 + 1)$, then $x^G \cap \Delta \neq \varnothing$ and $x^G \cap \Delta^c \neq \varnothing$.
\end{prop}

\begin{proof}
 This follows immediately by applying Lemma \ref{lem:main2} both to $\Gamma$ and its complement.
\end{proof}

\begin{cor}
 \label{cor:center}
 If $G$ is a group of order $v$ with a nontrivial center such that $\nu_2 - \nu_3$ divides neither $\mu - \nu_3(\nu_2 + 1)$ nor $v - 2k + \lambda - \nu_3( \nu_2 + 1)$, then $G$ cannot have a $(v,k,\lambda, \mu)$-PDS, i.e., $G$ cannot act regularly on the vertices of a non-conference $(v,k,\lambda, \mu)$-strongly regular graph.
\end{cor}

\begin{proof}
 If $z$ is a nontrivial element of $Z(G)$, then $z^G = \{z\}$.  If $G$ acts regularly on the vertices of a non-conference $(v,k,\lambda, \mu)$-strongly regular graph, by Proposition \ref{prop:comp}, $z^G \cap \Delta \neq \varnothing$ and $z^G \cap \Delta^c \neq \varnothing$, which is impossible.
\end{proof}

\begin{cor}
 \label{cor:pgroup}
 Let $p$ be a prime and $G$ be a nontrivial $p$-group.  If $\nu_2 - \nu_3$ divides neither $\mu - \nu_3(\nu_2 + 1)$ nor $v - 2k + \lambda - \nu_3(\nu_2 + 1)$, then there does not exist a $(v,k,\lambda, \mu)$-PDS in $G$, i.e., $G$ cannot act regularly on the vertices of a non-conference $(v,k,\lambda, \mu)$-strongly regular graph.
\end{cor}

\begin{proof}
 This follows from Corollary \ref{cor:center} and the fact that nontrivial $p$-groups have nontrivial centers.
\end{proof}

We note that the conditions of Corollaries \ref{cor:center} and \ref{cor:pgroup} are nontrivial, since there are many feasible parameter sets to which these conditions apply; see Section \ref{sect:infeas}.

\section{Infeasible parameters of partial difference sets in groups with a nontrivial center}
\label{sect:infeas}

Tables \ref{tbl:nonp} and \ref{tbl:p} provide instances when partial difference sets can be ruled out entirely, since in every instance a group of order $v$ must have a nontrivial center.  Note that $\nu_2 - \nu_3 = \sqrt{\Lambda}$, and the column ``$m$'' represents the value of $\mu - \nu_3( \nu_2 + 1)$.  Note that calculating $v - 2k + \lambda - \nu_3(\nu_2 + 1)$ for a $(v,k,\lambda, \mu)$-SRG $\Gamma$ is equivalent to calculating $m = \mu - \nu_3( \nu_2 + 1)$ for the complement of $\Gamma$.

\begin{table}[H]\centering
\begin{minipage}{0.4\textwidth}\centering
 {\footnotesize \begin{tabular}{cccccccc}
  $v$ & $k$ & $\lambda$ & $\mu$ & $\nu_2$ & $\nu_3$ & $\sqrt{\Lambda}$ & $m$ \\
  \hline
  28  & 12 & 6 & 4 & 4 & -2 & 6 & 14 \\
      & 15 & 6 & 10 & 1 & -5 & 6 & 20 \\
  63  & 30 & 13 & 15 & 3 & -5 & 8 & 35 \\
      & 32 & 16 & 16 & 4 & -4 & 8 & 36 \\
  88  & 27 & 6 & 9 & 3 & -6 & 9 & 33 \\
      & 60 & 41 & 40 & 5 & -4 & 9 & 64 \\
  105 & 26 & 13 & 4 & 11 & -2 & 13 & 28 \\
      & 78 & 55 & 66 & 1 & -12 & 13 & 90 \\
  105 & 32 & 4 & 12 & 2 & -10 & 12 & 42 \\
      & 72 & 51 & 45 & 9 & -3 & 12 & 75 \\
  105 & 52 & 21 & 30 & 2 & -11 & 13 & 63 \\
      & 52 & 29 & 22 & 10 & -3 & 13 & 55 \\
  117 & 36 & 15 & 9 & 9 & -3 & 12 & 39 \\
      & 80 & 52 & 60 & 2 & -10 & 12 & 90 \\
  176 & 25 & 0 & 4 & 3 & -7 & 10 & 32 \\
      & 150 & 128 & 126 & 6 & -4 & 10 & 154 \\
  176 & 45 & 18 & 9 & 12 & -3 & 15 & 48 \\
      & 130 & 93 & 104 & 2 & -13 & 15 & 143 \\
  176 & 70 & 18 & 34 & 2 & -18 & 20 & 88 \\
      & 105 & 68 & 54 & 17 & -3 & 20 & 108 \\
  176 & 70 & 24 & 30 & 4 & -10 & 14 & 80 \\
      & 105 & 64 & 60 & 9 & -5 & 14 & 110 \\
  189 & 48 & 12 & 12 & 6 & -6 & 12 & 54 \\
      & 140 & 103 & 105 & 5 & -7 & 12 & 147 \\
  195 & 96 & 46 & 48 & 6 & -8 & 14 & 104 \\
      & 98 & 49 & 49 & 7 & -7 & 14 & 105 \\
  208 & 75 & 30 & 25 & 10 & -5 & 15 & 80 \\
      & 132 & 81 & 88 & 4 & -11 & 15 & 143 \\
  \end{tabular}}
\end{minipage}\qquad
\begin{minipage}{0.4\textwidth}\centering
 {\footnotesize \begin{tabular}{cccccccc}
  $v$ & $k$ & $\lambda$ & $\mu$ & $\nu_2$ & $\nu_3$ & $\sqrt{\Lambda}$ & $m$ \\
  \hline
  208 & 81 & 24 & 36 & 3 & -15 & 18 & 88 \\
      & 126 & 80 & 70 & 14 & -4 & 18 & 130 \\
  225 & 96 & 51 & 33 & 21 & -3 & 24 & 99 \\
      & 128 & 64 & 84 & 2 & -22 & 24 & 150 \\
  231 & 30 & 9 & 3 & 9 & -3 & 12 & 33 \\
      & 200 & 172 & 180 & 2 & -10 & 12 & 210 \\
  231 & 40 & 20 & 4 & 18 & -2 & 20 & 42 \\
      & 190 & 153 & 171 & 1 & -19 & 20 & 209 \\
  231 & 90 & 33 & 36 & 6 & -9 & 15 & 99 \\
      & 140 & 85 & 84 & 8 & -7 & 15 & 147 \\
  232 & 33 & 2 & 5 & 4 & -7 & 11 & 40 \\
      & 198 & 169 & 168 & 6 & -5 & 11 & 203 \\
  232 & 63 & 14 & 18 & 5 & -9 & 14 & 72 \\
      & 168 & 122 & 120 & 8 & -6 & 14 & 174 \\
  232 & 77 & 36 & 20 & 19 & -3 & 22 & 80 \\
      & 154 & 96 & 114 & 2 & -20 & 22 & 174 \\
   232 & 81 & 30 & 27 & 9 & -6 & 15 & 87 \\
       & 150 & 95 & 100 & 5 & -10 & 15 & 160 \\
   236 & 55 & 18 & 11 & 11 & -4 & 15 & 59 \\
       & 180 & 135 & 144 & 3 & -12 & 15 & 192 \\
   275 & 112 & 30 & 56 & 2 & -28 & 30 & 140 \\
       & 162 & 105 & 81 & 27 & -3 & 30 & 165 \\
   279 & 128 & 52 & 64 & 4 & -16 & 20 & 144 \\
       & 150 & 85 & 75 & 15 & -5 & 20 & 155 \\
   285 & 64 & 8 & 16 & 4 & -12 & 17 & 76 \\
       & 220 & 171 & 165 & 11 & -5 & 16 & 225 \\
   297 & 128 & 64 & 48 & 20 & -4 & 24 & 132 \\
       & 168 & 87 & 105 & 3 & -21 & 24 & 189 \\
  \end{tabular}}
\end{minipage}
\caption{Parameters with $v \le 300$ ruled out by Corollary \ref{cor:center}}
\label{tbl:nonp}
\end{table}

\begin{table}[H]
 {\footnotesize \begin{tabular}{cccccccc}
  $v$ & $k$ & $\lambda$ & $\mu$ & $\nu_2$ & $\nu_3$ & $\sqrt{\Lambda}$ & $m$ \\
  \hline
  343 & 102 & 21 & 34 & 4 & -17 & 21 & 119 \\
      & 240 & 171 & 160 & 16 & -5 & 21 & 245 \\
  343 & 114 & 45 & 34 & 16 & -5 & 21 & 119 \\
      & 228 & 147 & 160 & 4 & -17 & 21 & 245 \\
  625 & 246 & 119 & 82 & 41 & -4 & 45 & 250 \\
      & 378 & 213 & 252 & 3 & -42 & 45 & 420 \\
  729 & 208 & 37 & 68 & 4 & -35 & 39 & 243 \\
      & 520 & 379 & 350 & 34 & -5 & 29 & 525 \\
  729 & 248 & 67 & 93 & 5 & -31 & 36 & 279 \\
      & 480 & 324 & 300 & 30 & -6 & 36 & 486 \\
  729 & 280 & 127 & 95 & 37 & -5 & 42 & 285 \\
      & 448 & 262 & 296 & 4 & -38 & 42 & 486 \\
  \end{tabular}}
 \caption{Parameters with $v \le 1000$ ruled out by Corollary \ref{cor:pgroup}}
 \label{tbl:p}
\end{table}

Tables \ref{tbl:center1} and \ref{tbl:center2} provide instances when partial difference sets can be ruled out for groups of order $v$ with nontrivial centers.  Note that $\nu_2 - \nu_3 = \sqrt{\Lambda}$, and the column ``$m$'' represents the value of $\mu - \nu_3( \nu_2 + 1)$.

\begin{table}[H]
\begin{minipage}{0.4\textwidth}\centering
 {\footnotesize \begin{tabular}{cccccccc}
  $v$ & $k$ & $\lambda$ & $\mu$ & $\nu_2$ & $\nu_3$ & $\sqrt{\Lambda}$ & $m$ \\
  \hline
  36  & 14 & 7 & 4 & 5 & -2 & 7 & 16  \\
      & 21 & 10 & 15 & 1 & -6 & 7 & 27 \\
  50  & 21 & 8 & 9 & 3 & -4 & 7 & 25 \\
      & 28 & 15 & 16 & 3 & -4 & 7 & 32 \\
  66  & 20 & 10 & 4 & 8 & -2 & 10 & 22 \\
      & 45 & 28 & 36 & 1 & -9 & 10 & 54 \\
  70  & 27 & 12 & 9 & 6 & -3 & 9 & 30 \\
      & 42 & 23 & 28 & 2 & -7 & 9 & 49 \\
  78  & 22 & 11 & 4 & 9 & -2 & 11 & 24 \\
      & 55 & 36 & 45 & 1 & -10 & 11 & 65 \\
  96  & 35 & 10 & 14 & 3 & -7 & 10 & 42 \\
      & 60 & 38 & 36 & 6 & -4 & 10 & 64 \\
  100 & 33 & 14 & 9 & 8 & -3 & 11 & 36 \\
      & 66 & 41 & 48 & 2 & -9 & 11 & 75 \\
  120 & 28 & 14 & 4 & 12 & -2 & 14 & 30 \\
      & 91 & 66 & 78 & 1 & -13 & 14 & 104 \\
  120 & 42 & 8 & 18 & 2 & -12 & 14 & 54 \\
      & 77 & 52 & 44 & 11 & -3 & 14 & 80 \\
  126 & 25 & 8 & 4 & 7 & -3 & 10 & 28 \\
      & 100 & 78 & 84 & 2 & -8 & 10 & 108 \\
  126 & 50 & 13 & 24 & 2 & -13 & 15 & 63 \\
      & 75 & 48 & 39 & 12 & -3 & 15 & 78 \\
  126 & 60 & 33 & 24 & 12 & -3 & 15 & 63 \\
      & 65 & 28 & 39 & 2 & -13 & 15 & 78 \\
  130 & 48 & 20 & 16 & 8 & -4 & 12 & 52 \\
      & 81 & 48 & 54 & 3 & -9 & 12 & 90 \\
  136 & 30 & 8 & 6 & 6 & -4 & 10 & 34 \\
      & 105 & 80 & 84 & 3 & -7 & 10 & 112 \\
  136 & 60 & 24 & 28 & 4 & -8 & 12 & 68 \\
      & 75 & 42 & 40 & 7 & -5 & 12 & 80 \\
  136 & 63 & 30 & 28 & 7 & -5 & 12 & 68 \\
      & 72 & 36 & 40 & 4 & -8 & 12 & 80 \\
  148 & 63 & 22 & 30 & 3 & -11 & 14 & 74 \\
      & 84 & 50 & 44 & 10 & -4 & 14 & 88 \\
  148 & 70 & 36 & 30 & 10 & -4 & 14 & 74 \\
      & 77 & 36 & 44 & 3 & -11 & 14 & 88 \\
  154 & 48 & 12 & 16 & 4 & -8 & 12 & 56 \\
      & 105 & 72 & 70 & 7 & -5 & 12 & 110 \\
  154 & 72 & 26 & 40 & 2 & -16 & 18 & 88 \\
      & 81 & 48 & 36 & 15 & -3 & 18 & 84 \\
  170 & 78 & 35 & 36 & 6 & -7 & 13 & 85 \\
      & 91 & 48 & 49 & 6 & -7 & 13 & 98 \\
  171 & 34 & 17 & 4 & 15 & -2 & 17 & 36 \\
      & 136 & 105 & 120 & 1 & -16 & 17 & 152 \\
  171 & 50 & 13 & 15 & 5 & -7 & 12 & 57 \\
      & 120 & 84 & 84 & 6 & -6 & 12 & 126 \\
  171 & 60 & 15 & 24 & 3 & -12 & 15 & 72 \\
      & 110 & 73 & 66 & 11 & -4 & 15 & 114 \\
  \end{tabular}}
\end{minipage}\qquad
\begin{minipage}{0.4\textwidth}\centering
 {\footnotesize \begin{tabular}{cccccccc}
  $v$ & $k$ & $\lambda$ & $\mu$ & $\nu_2$ & $\nu_3$ & $\sqrt{\Lambda}$ & $m$ \\
  \hline
  190 & 36 & 18 & 4 & 16 & -2 & 18 & 38 \\
      & 153 & 120 & 136 & 1 & -17 & 18 & 170 \\
  190 & 45 & 12 & 10 & 7 & -5 & 12 & 50 \\
      & 144 & 108 & 112 & 4 & -8 & 12 & 152 \\
  190 & 84 & 33 & 40 & 4 & -11 & 15 & 95 \\
      & 105 & 60 & 55 & 10 & -5 & 15 & 110 \\
  190 & 84 & 38 & 36 & 8 & -6 & 14 & 90 \\
      & 105 & 56 & 60 & 5 & -9 & 14 & 114 \\
  190 & 90 & 45 & 40 & 10 & -5 & 15 & 95 \\
      & 99 & 48 & 55 & 4 & -11 & 15 & 110 \\
  196 & 39 & 2 & 9 & 3 & -10 & 13 & 49 \\
      & 156 & 125 & 120 & 9 & -4 & 13 & 160 \\
  196 & 60 & 23 & 16 & 11 & -4 & 15 & 64 \\
      & 135 & 90 & 99 & 3 & -12 & 15 & 147 \\
  204 & 63 & 22 & 18 & 9 & -5 & 14 & 68 \\
      & 140 & 94 & 100 & 4 & -10 & 14 & 150 \\
  210 & 38 & 19 & 4 & 17 & -2 & 19 & 40 \\
      & 171 & 136 & 153 & 1 & -18 & 19 & 189 \\
  220 & 84 & 38 & 28 & 14 & -4 & 18 & 88 \\
      & 135 & 78 & 90 & 3 & -15 & 18 & 150 \\
  222 & 51 & 20 & 9 & 14 & -3 & 17 & 54 \\
      & 170 & 127 & 140 & 2 & -15 & 17 & 185 \\
  238 & 75 & 20 & 25 & 5 & -10 & 15 & 85 \\
      & 162 & 111 & 108 & 9 & -6 & 15 & 168 \\
  244 & 108 & 42 & 52 & 4 & -14 & 18 & 121 \\
      & 135 & 78 & 70 & 13 & -5 & 18 & 140 \\
  244 & 117 & 60 & 52 & 13 & -5 & 18 & 122 \\
      & 126 & 60 & 70 & 4 & -14 & 18 & 140 \\
  246 & 85 & 20 & 34 & 3 & -17 & 20 & 102 \\
      & 160 & 108 & 96 & 16 & -4 & 20 & 164 \\
  246 & 105 & 36 & 51 & 3 & -18 & 21 & 123 \\
      & 140 & 85 & 72 & 17 & -4 & 21 & 144 \\
  246 & 119 & 64 & 51 & 17 & -4 & 21 & 123 \\
      & 126 & 57 & 72 & 3 & -18 & 21 & 144 \\
  266 & 45 & 0 & 9 & 3 & -12 & 15 & 57 \\
      & 220 & 183 & 176 & 11 & -4 & 15 & 224 \\
  273 & 80 & 19 & 25 & 5 & -11 & 16 & 91 \\
      & 192 & 136 & 132 & 10 & -6 & 16 & 198 \\
  273 & 102 & 41 & 36 & 11 & -6 & 17 & 108 \\
      & 170 & 103 & 110 & 5 & -12 & 17 & 182 \\
  273 & 136 & 65 & 70 & 6 & -11 & 17 & 147 \\
      & 136 & 69 & 66 & 10 & -7 & 17 & 143 \\
  276 & 44 & 22 & 4 & 20 & -2 & 22 & 46 \\
      & 231 & 190 & 210 & 1 & -21 & 22 & 252 \\
  276 & 75 & 10 & 24 & 3 & -17 & 20 & 92 \\
      & 200 & 148 & 136 & 16 & -4 & 20 & 204 \\
  276 & 75 & 18 & 21 & 6 & -9 & 15 & 84 \\
      & 200 & 145 & 144 & 8 & -7 & 15 & 207 \\
\end{tabular}}
\end{minipage}
\caption{Parameters ruled out for groups with a nontrivial center when $v \le 300$ by Corollary \ref{cor:center}}
\label{tbl:center1}
\end{table}

\begin{table}[H]
{\footnotesize \begin{tabular}{cccccccc}
  $v$ & $k$ & $\lambda$ & $\mu$ & $\nu_2$ & $\nu_3$ & $\sqrt{\Lambda}$ & $m$ \\
  \hline
  276 & 110 & 52 & 38 & 18 & -4 & 22 & 114 \\
      & 165 & 92 & 108 & 3 & -19 & 22 & 184 \\
  276 & 135 & 78 & 54 & 27 & -3 & 30 & 138 \\
      & 140 & 58 & 84 & 2 & -28 & 30 & 168 \\
  280 & 117 & 44 & 52 & 5 & -13 & 18 & 130 \\
      & 162 & 96 & 90 & 12 & -6 & 18 & 168 \\
  286 & 95 & 24 & 33 & 4 & -15 & 19 & 108 \\
      & 190 & 129 & 120 & 14 & -5 & 19 & 195 \\
  286 & 125 & 60 & 50 & 15 & -5 & 20 & 130 \\
      & 160 & 84 & 96 & 4 & -16 & 20 & 176 \\
  288 & 105 & 52 & 30 & 25 & -3 & 28 & 108 \\
      & 182 & 106 & 130 & 2 & -26 & 28 & 208 \\
  290 & 136 & 63 & 64 & 8 & -9 & 17 & 145 \\
      & 153 & 80 & 81 & 8 & -9 & 17 & 161 \\
  300 & 46 & 23 & 4 & 21 & -2 & 23 & 48 \\
      & 253 & 210 & 231 & 1 & -22 & 23 & 275 \\
\end{tabular}}
\caption{Parameters ruled out for groups with a nontrivial center when $v \le 300$ by Corollary \ref{cor:center}, continued}
\label{tbl:center2}
\end{table}

\noindent\textsc{Acknowledgments.}
The authors wish to thank the anonymous referees for their helpful reports.

\bibliographystyle{plain}
\bibliography{PDS}

\end{document}

config: arxiv


TITLE: Spaces of class $J_\alpha$
QUESTION [2 upvotes]: This question is about the spaces of class $J_\alpha$.
Given three Banach spaces $Z\subset Y\subset X$ (with continuous embeddings), and given $\alpha\in (0,1)$, we say that $Y$ is of class $J_\alpha$ between $X$ and $Z$ if there is $C>0$ such that
\begin{equation}
\left\|y\right\|_Y\leq C \left\|y\right\|_Z^\alpha\left\|y\right\|_X^{1-\alpha}, \quad y\in Z.
\end{equation}
My question is:  Can we say that the space $C[0,1]$ is of class $J_\alpha$ between
$H^2(0,1)$ and $L^2(0,1)$.
It is obvious that $H^2(0,1)\subset C[0,1] \subset L^2(0,1)$. In addition, one can show that there is $\tilde{C}>0$ such that
\begin{equation}
\left\|y\right\|_{\infty}\leq \tilde{C} \left\|y\right\|_{H^2},
\end{equation}
which in a way reflects the embedding $H^2(0,l) \subset C[0,l]$.
Also, we trivially have
\begin{equation}
\left\|y\right\|_{L^2}\leq \left\|y\right\|_{\infty}.
\end{equation}
I am not sure if the original assumption is correct and would be really grateful for any suggestions!

REPLY [1 votes]: One approach is to integrate by parts. This works nicely when $y(0)=y(1)=0$, because then
$$\int_0^1 (y')^2 =-\int_0^1 y\,y''\le \|y\|_{L^2}\|y''\|_{L^2}\le \|y\|_{L^2}\|y\|_{H^2} \tag1$$
which implies
$$\|y\|_{L^\infty} \le \|y'\|_{L^1} \le \|y'\|_{L^2} \le \|y\|_{L^2}^{1/2}\|y\|_{H^2}^{1/2} \tag2$$
In general one has to deal with boundary terms, which is possible but less pleasant.

"you should totally drop that and try jQuery Fourier series"

Indeed, let's write $y(t)=\sum c_n e^{2\pi int}$; here and below summation is over ${n\in\mathbb Z}$. The norms are roughly $\|y\|_{L^2}=(\sum |c_n|^2)^{1/2}$ and $\|y\|_{H^2}=(\sum (|n|+1)^4 |c_n|^2)^{1/2}$. Clearly, $\|y\|_\infty\le \sum |c_n|$. The problem reduces to  the numerology of  exponents in Hölder's inequality (a.k.a. Hölderology).  For example,
$$\sum |c_n| = \sum |c_n|^{1/2}|c_n|^{1/2} \le \left(\sum |c_n|^2\right)^{1/4} \left(\sum |c_n|^{2/3}\right)^{3/4} \tag3$$
and then
$$\sum |c_n|^{2/3} = \sum  (|n|+1)^{-4/3} (|n|+1)^{4/3}|c_n|^{2/3}
\\ \\ \le \left(\sum  (|n|+1)^{-2}\right)^{2/3} \left( \sum (|n|+1)^{4}|c_n|^{2}  \right)^{1/3}
\tag4$$
The end result is
$$\begin{split}\sum |c_n| &\le \left(\sum |c_n|^2\right)^{1/4} \left(\sum  (|n|+1)^{-2}\right)^{1/2} \left( \sum (|n|+1)^{4}|c_n|^{2}  \right)^{1/4} \\
&\le C\|y\|_{L^2}^{1/2}\|y\|_{H^2}^{1/2} \end{split} \tag5$$

There is some slack in (5): it uses $\sum(|n|+1)^{-2}<\infty$, but $\sum  (|n|+1)^{-r}$ converges for any $r>1$. You may want to redo the Hölderology to find  $\alpha<1/2$ for which $\sum |c_n| \le C\|y\|_{L^2}^{1-\alpha}\|y\|_{H^2}^{\alpha}$ holds. Can you find the smallest possible $\alpha$?
Here is a lower bound for $\alpha$, by means of the example
$y(t)=((\epsilon-t)^+)^2$. Here $\|y\|_\infty = \epsilon^2$, $\|y\|_{L^2}\approx \epsilon^{5/2}$ and $\|y\|_{H^2}\approx \epsilon^{1/2}$. Therefore,
$$\epsilon^2 \le C \epsilon^{\alpha/2}\epsilon^{5(1-\alpha)/2}$$
for arbitrarily small $\epsilon$, which implies $\alpha\ge 1/4$.