TITLE: Lebesgue Integration of Non-negative functions
QUESTION [0 upvotes]: We define the Lebesgue integral of non-negative functions as:
$$\int f(x)dx=\sup_g\int g(x)dx$$
where the sup is taken over all measurable functions $g$ such that $0\leq g\leq f$ and $g$ bounded, $m(supp(g))<\infty$.
My Question:

In this definition, is it inherent that the Lebesgue integral is approximating the volume of a function from "below" (the condition that $0\leq g \leq f)$?
Is this why in a sense the Lebesgue integral covers a larger class of functions than the Riemann since for the Riemann, we need both the upper sum and lower sum to coincide, which loosely speaking, is tantamount to approximating the volume from both above and below?

REPLY [1 votes]: Actually there is a standard theorem in real analysis saying that, for a bounded domain $D$ and bounded function $f$,
\begin{align*}
\sup_{\varphi\leq f}\int_{D}\varphi=\inf_{\psi\geq f}\int_{D}\psi,
\end{align*}
the running variables $\varphi,\psi$ are all simple functions. You may figure out this theorem in, for example, Royden's book. In view of this theorem, it still gives you some sort of interpretation that one can also approximate the integral of bounded $f$ under bounded domain from above.
I think the crucial difference is that one really does the partition of the $y$-axis rather than the $x$-axis which the latter is the essence of Riemann integrability. Now assume that $f[0,1]\subseteq[0,1]$ and we are dealing with the integral on the bounded domain $[0,1]$, and the range of $f$ lies in $[0,1]$. Partitioning through $y$-axis gives you that,
\begin{align*}
\sum_{i=1}^{n}\dfrac{i}{n}\chi_{f^{-1}[(i-1)/n,i/n]}\geq f,
\end{align*}
those $f^{-1}[(i-1)/n,i/n]$ are in some sense of the grids on the $y$-axis because one does the partition of the range of $f$, in this case, which is $[0,1]$ into $[(i-1)/n,i/n]$ and pulls back to the $x$-axis under the operation $f^{-1}$.


TITLE: Limit of $f_{n+1} = \sqrt{12 + f_n}$ with proof by contradiction
QUESTION [4 upvotes]: Consider the following recursive sequence:
$$
\begin{cases}
f_{0}=\sqrt{12}\\
f_{n+1}=\sqrt{12 + f_{n}}
\end{cases}
$$
for $n \geq 0$. How can I prove that this sequence is bounded above by $4$ and
thus its limit is also $4$?

REPLY [0 votes]: Hint:
To prove a recursive sequence is convergent, one strategy is to prove it is non-decreasing and bounded from above, or its non-increasing and bounded from below.
If the sequence is defined with a function $g$,: $f_{n+1}=g(f_n)$, it is  non-decreasing if the graph of $g$ is above  the line $y=x$ on an interval $I$ and all $f_n$ belong to $I$. If the interval $I$ is bounded from above, the sequence will also be bounded from above, hence it will converge.
Furthermore, if $g$ is continuous, the limit $\ell$ will satisfy the equation $\; \ell=g(\ell)$, in other words, the limit will be a fixed point of $g$. If there is only one fixed point, it will be the limit. If there are several fixed  points, you will have to decide (and prove) which fixed point is the limit.


text:
\begin{document}

\title[Natural Gradient for Combined Loss]{Natural Gradient for Combined Loss Using Wavelets}

\author[]{Lexing Ying}
\address[Lexing Ying]{Department of Mathematics and ICME, Stanford University, Stanford, CA 94305}
\email{lexing@stanford.edu}

\thanks{The work of L.Y. is partially supported by the U.S. Department of Energy, Office of Science,
  Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing
  (SciDAC) program and also by the National Science Foundation under award DMS-1818449.
}

\keywords{Natural gradient, Fisher-Rao metric, Wasserstein metric, Mahalanobis metric, compactly
  supported wavelet, diagonal approximation.}

\begin{abstract}
  Natural gradients have been widely used in optimization of loss functionals over probability
  space, with important examples such as Fisher-Rao gradient descent for Kullback-Leibler
  divergence, Wasserstein gradient descent for transport-related functionals, and Mahalanobis
  gradient descent for quadratic loss functionals. This note considers the situation in which the
  loss is a convex linear combination of these examples. We propose a new natural gradient algorithm
  by utilizing compactly supported wavelets to diagonalize approximately the Hessian of the combined
  loss. Numerical results are included to demonstrate the efficiency of the proposed algorithm.
\end{abstract}

\maketitle

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

Many problems in partial differential equations and machine learning can be formulated as
optimization problems over probability densities. For a domain $\Omega$, let $E(p)$ be a loss or
energy functional defined for the probability densities $p$ over $\Omega$. The goal is to find $p^*$
that minimizes $E(p)$. A common approach, especially for $E(p)$ with a unique minimum, is to follow
the gradient descent (GD) dynamics. However, depending the metric used in the gradient calculation,
different gradient descent algorithms exhibit drastically different convergence behavior. The term
{\em natural gradient} refers to the practice of choosing an appropriate metric depending on the
loss functional $E(p)$ as well as the probability space. Below are several well-known examples of
natural gradient.
\begin{itemize}
\item Wasserstein GD that scales the Euclidean gradient $\frac{\delta E}{\delta p}(p)$ with the
  metric $-\nabla\cdot(p\nabla)$. Wasserstein GD is typically effective for a loss $E(p)$ that
  behaves like the square of the 2nd Wasserstein distance.
\item Fisher-Rao GD that scales the Euclidean gradient  $\frac{\delta E}{\delta p}(p)$ with the diagonal
  tensor $\diag(p)$. Fisher-Rao GD is quite effective for a loss $E(p)$ such as the Kullback-Leibler
  divergence $\int p(x) \ln\frac{p(x)}{\mu(x)} \d x$.
\item Mahalanobis GD that scales the Euclidean gradient $\frac{\delta E}{\delta p}(p)$ with a positive
  definite metric $B$. Mahalanobis GD is efficient for a quadratic loss of the form
  $\frac{1}{2}(p-\mu, A(p-\mu))$ with $B \approx A^{-1}$. In this note, we consider the case that $A$
  is a positive semidefinite pseudo-differential operator, for example $A=-\Delta$.
\end{itemize}
A general principle from these examples is that, for a natural gradient to be effective, the metric
used at the density $p$ should be an approximate inverse of the Hessian of the loss $E(p)$ at
$p$. In each of these three examples, an approximate inverse of the Hessian can be derived quite
explicitly.

\subsection{Problem statement.}

In several problems from kinetic theory and statistical machine learning, one is faced with a loss
or energy functional $E(p)$ that is a linear combination of these three forms mentioned above, i.e.,
\[
E(p) = \alpha_1 E_1(p) + \alpha_2 E_2(p) + \alpha_3 E_3(p),
\]
where $\alpha_1, \alpha_2, \alpha_3 \ge 0$ and $E_1$, $E_2$, and $E_3$ are of the Wasserstein,
Fisher-Rao, and Mahalanobis types, respectively, i.e.,
\[
\frac{\delta^2 E_1}{\delta p^2}(p) \approx (-\nabla\cdot(p\nabla))^+,\quad
\frac{\delta^2 E_2}{\delta p^2}(p) \approx \diag\left(\frac{1}{p}\right), \quad
\frac{\delta^2 E_3}{\delta p^2}(p) \approx A
\]
where $(\cdot)^+$ stands for pseudo-inverse. As a result, the Hessian of $E(p)$ has the following
approximation
\[
\frac{\delta^2 E}{\delta p^2}(p) =
\alpha_1 \frac{\delta^2 E_1}{\delta p^2}(p) +
\alpha_2 \frac{\delta^2 E_2}{\delta p^2}(p) +
\alpha_3 \frac{\delta^2 E_3}{\delta p^2}(p).
\]
None of three natural gradients listed above is effective for this combined loss functional, since
the inverse of $\frac{\delta^2 E}{\delta p^2}(p)$ looks quite different from
$-\nabla\cdot(p\nabla)$, $\diag(p)$, or $A^{-1}$.

An immediate question is design an efficient natural gradient (or even an approximate one) for the
combined loss $E(p)$. Due to the efficiency considerations, we prefer this natural gradient to have
the following features.
\begin{itemize}
\item It utilizes the Hessian information of $E_1(p)$, $E_2(p)$, and $E_3(p)$ in the design of the
  natural gradient.
\item It avoids forming and/or inverting the Hessian $\frac{\delta^2 E}{\delta p^2}(p)$ in order to
  avoid super-linear costs.
\item The computational cost of computing the natural gradient from $\frac{\delta E}{\delta p}(p)$
  should be of order $O(n\log^c n)$, where $n$ is the number of degrees of freedom used for
  discretizing $p$.
\end{itemize}

The main idea of our approach is to diagonalize $\frac{\delta^2 E}{\delta p^2}(p)$ approximately by
finding a common basis that diagonalizes each of the three terms $\frac{\delta^2 E_1}{\delta
  p^2}(p)$, $\frac{\delta^2 E_2}{\delta p^2}(p)$, and $\frac{\delta^2 E_3}{\delta p^2}(p)$
approximately at the same time. Among various choices, compactly supported wavelets emerge as a
natural candidate because they approximately diagonalize (1) differential operators, (2) diagonal
scaling by functions with sufficient regularity, and also (3) pseudo-differential operators.

\subsection{Related work.}

Fisher-Rao metric is essential to many branches of probability and statistics, as it is invariant
under diffeomorphisms. The study of Fishe-Rao and related metrics has evolved to become the field of
information geometry and we refer to \cite{amari2016information,ay2017information} for detail
discussions. Explicit time-discretization of the Fisher-Rao GD gives rise the mirror descent
algorithms \cite{beck2003mirror,bubeck2015convex,nemirovsky1983problem}, which plays an essential
role in online learning and optimization.

Originated from the theory of optimal transport, Wasserstein metric is defined formally as the
Hessian of the square of the 2nd Wasserstein distance
\cite{villani2003topics,villani2008optimal,santambrogio2015optimal,peyre2019computational}. Starting
from \cite{jordan1998variational,otto2001geometry}, it has been shown that many kinetic-type PDEs
can be viewed as a Wasserstein GD of free energies defined on probability spaces
\cite{carrillo2003kinetic}. In recent years, a parametric version of the Wasserstein metric has been
applied to various applied problems from statistical machine learning
\cite{chen2018natural,li2018natural}.

The quadratic term associated with the Mahalanobis metric appears quite often in partial
differential equation models, for example as the Dirichlet energy or as the interacting free energy
term in the Keller-Segel models \cite{Perthame2006transport}.

A recent paper \cite{ying2020mirror} considers the case where the loss function is the sum of the
Kullback-Leibler divergence and a quadratic interacting term. By adopting a diagonal approximation
of interacting term, it proposes new natural gradient dynamics and develops new mirror descent
algorithms.

\subsection{Contents.}
The rest of this note is organized as follows. Section \ref{sec:algo} proposes a new metric for the
combined loss functional and derives the natural gradient algorithm. In Section \ref{sec:res},
numerical results in 1D and 2D show that the proposed natural gradient outperforms the existing ones
for combined loss functionals. Finally, Section \ref{sec:disc} ends with some discussions on future
work.

\subsection{Data availability statement.}
Data sharing not applicable to this article as no datasets were generated or analyzed during the
current study.

\section{Algorithm} \label{sec:algo}

\subsection{Metric design} \label{sec:metric}

Consider the 1D problem with $\Omega=[0,1]$ with the periodic boundary condition for simplicity. As
mentioned above for the loss functional $E(p)=\alpha_1 E_1(p)+ \alpha_2 E_2(p)+ \alpha_3 E_3(p)$,
the Hessian can be approximated as follows.
\begin{equation} \label{eq:ddEddp}
  \frac{\delta^2 E}{\delta p^2}(p) =
  \alpha_1\frac{\delta^2 E_1}{\delta p^2}(p) +
  \alpha_2\frac{\delta^2 E_2}{\delta p^2}(p) +
  \alpha_3\frac{\delta^2 E_3}{\delta p^2}(p)
  \approx
  \alpha_1
(-\nabla\cdot(p\nabla))^+ + \alpha_2 \diag\left(\frac{1}{p}\right) + \alpha_3 A.
\end{equation}
For simplicity, assume that the domain $\Omega$ is discretized with a uniform grid with $n$ points
$S = \left\{\frac{0}{n},\frac{1}{n},\ldots,\frac{n-1}{n}\right\}$. A density $p(x)$ for
$x\in\Omega=[0,1]$ can be represented as a vector $p\in\R^n$ with entries denoted by $p_s$ for $s\in
S$.  We denote by $D$ the discrete differential operator. After the discretization, the Hessian
approximation \eqref{eq:ddEddp} takes the following discrete form
\begin{equation} \label{eq:Eppdisc}
  \frac{\delta^2 E}{\delta p^2}(p) \approx \alpha_1 \left(D^\T\diag(p)D\right)^+ + \alpha_2
  \diag\left(\frac{1}{p}\right) + \alpha_3 A.
\end{equation}

As mentioned earlier, the key idea is to diagonalize each of the three terms in \eqref{eq:Eppdisc}
in a compactly supported orthogonal wavelet basis such as the Daubechies wavelets
\cite{daubechies1992ten,mallat1999wavelet}. Let us denote by $W\in \R^{n\times n}$ the matrix such
that its $j$-th column is the $j$-th vector of the wavelet basis. Therefore, $W$ is the matrix for
wavelet reconstruction and its transpose $W^\T$ is the matrix for wavelet decomposition. Notice that
for compactly supported wavelets, $W$ and $W^\T$ are sparse matrices with only $O(n\log n)$ non-zero
entries. Applying $W$ or $W^\T$ to an arbitrary vector of length $n$ takes only $O(n)$ operations by
taking advantages of the filter bank structure of the wavelet basis \cite{mallat1999wavelet}.

Applying the matrices $W^\T$ to the left and $W$ to the right of \eqref{eq:Eppdisc} leads to
\[
W^\T \frac{\delta^2 E}{\delta p^2}(p) W
\approx
\alpha_1 W^\T \left(D^\T\diag(p)D\right)^+ W +
\alpha_2 W^\T \diag\left(\frac{1}{p}\right) W +
\alpha_3 W^\T A W.
\]
The three terms on the right hand side are treated as follows.
\begin{itemize}
\item For the first term, consider first its pseudo-inverse $W^\T D^\T \diag(p) D W$. The
  diagonal entries of $W^\T D^\T\diag(p)DW$ at the $(i,i)$ slot is given by
  \[
  \sum_{s\in S} (DW)_{si} p_s (DW)_{si} = \sum_{s\in S} (DW)_{si}^2 p_s.
  \]
  By defining the matrix $H_1$ with entries given by $(H_1)_{is} = (DW)_{si}^2$, the whole diagonal
  of $W^\T D^\T\diag(p)DW$ can be conveniently written as $H_1 p$, which clearly depends linearly on
  $p$. Taking its pseudo-inverse implies that $W^\T\left(D^\T\diag(p)D\right)^+ W$ can be diagonally
  approximated with $\diag\left(\frac{1}{H_1 p}\right)$.

\item For the second term, consider first its pseudo-inverse $W^\T \diag(p) W$. The diagonal
  entry $W^\T \diag(p) W$ at the $(i,i)$ slot is given by
  \[
  \sum_{s\in S} W_{si} p_s W_{si} = \sum_{s\in S} W_{si}^2 p_s.
  \]
  By defining the matrix $H_2 p$ with entries given by $(H_1)_{is} = W_{si}^2$, the whole diagonal
  of $W^\T \diag(p) W$ can be written as $H_2 p$, which is again linear in $p$. Taking its
  pseudo-inverse shows that $W^\T \diag\left(\frac{1}{p}\right) W$ can be diagonally approximated
  with $\diag\left(\frac{1}{H_2 p}\right)$.
\item As opposed to the first two terms, the third term $W^\T A W$ is independent of the density
  $p$. Its diagonal can be precomputed and will be denoted by $h_3 \in \R^n$.
\end{itemize}

Putting the three terms together, we conclude that
\[
W^\T \frac{\delta^2 E}{\delta p^2}(p) W \approx
\diag\left(
\frac{\alpha_1}{H_1p} + \frac{\alpha_2}{H_2p} + \alpha_3 h_3
\right),
\]
or equivalently
\[
\frac{\delta^2 E}{\delta p^2}(p) \approx
W
\diag\left(
\frac{\alpha_1}{H_1p} + \frac{\alpha_2}{H_2p} + \alpha_3 h_3
\right)
W^\T.
\]
By inverting this approximation, we reach at the metric for the natural gradient
\begin{equation}\label{eqn:metric}
\left( \frac{\delta^2 E}{\delta p^2}(p) \right)^{-1} \approx
W
\diag
\left(
\frac{1}{
  \frac{\alpha_1}{H_1p} + \frac{\alpha_2}{H_2p} + \alpha_3 h_3
}
\right)
W^\T.
\end{equation}

With the metric ready, the ODE for the new natural gradient reads
\begin{equation}\label{eqn:pdot}
  \dot{p} = - W \diag \left( \frac{1}{\frac{\alpha_1}{H_1p} + \frac{\alpha_2}{H_2p} + \alpha_3 h_3}  \right) W^\T
  \frac{\delta E}{\delta p}(p).
\end{equation}
If we denote the wavelet coefficient vector by $c\in \R^n$, i.e., $c=W^\T p$ and $p = Wc$,
\eqref{eqn:pdot} can be written as
\begin{equation}\label{eqn:cdot}
  \dot{c} = -\diag \left( \frac{1}{\frac{\alpha_1}{H_1Wc} + \frac{\alpha_2}{H_2Wc} + \alpha_3 h_3} \right)
  \frac{\delta E}{\delta c}(c).
\end{equation}
In what follows, we simply refer to them as the {\em combined} gradient descent.

The following two claims show that the objects in \eqref{eqn:pdot} and \eqref{eqn:cdot} can be
computed efficiently.

\begin{claim}
  The computational cost of forming and storing the matrices $H_1$ and $H_2$ is $O(n\log n)$.
\end{claim}
\begin{proof}
  Let us recall the definition of the matrices $H_1$ and $H_2$
  \[
  (H_1)_{is} = (DW)_{si}^2,\quad
  (H_2)_{is} = W_{si}^2.
  \]
  Since the wavelets are compactly supported with a constant size support at the finest scale,
  applying the differential operator and taking the element-wise square for a wavelet at scale
  $\ell$ takes $O(n/2^\ell)$ steps. Summing over the wavelets from all scales gives the following
  bound for the total cost:
  \[
  \sum_{\ell=1}^{\log_2 n} 2^\ell \cdot \frac{n}{2^\ell} = O(n\log n).
  \]
\end{proof}

\begin{claim}
  For a density $p\in\R^n$ with $p_i >0$, the computational cost of applying the metric $W \diag
  \frac{1}{ \left( \frac{\alpha_1}{H_1p} + \frac{\alpha_2}{H_2p} + \alpha_3 h_3 \right) } W^\T$
  takes $O(n\log n)$ steps.
\end{claim}
\begin{proof}
  As a consequence from the previous claim, forming $H_1 p$ and $H_2 p$ each takes $O(n\log n)$
  steps. Applying the wavelet decomposition operator $W^\T$ or the reconstruction operator $W$ takes
  $O(n)$ steps by taking advantages of the filter bank construction. Summing them together gives the
  $O(n\log n)$ total cost.
\end{proof}

\subsection{Time discretization} \label{sec:algodetail}

Let us now describe the time discretization of the natural gradient dynamics \eqref{eqn:pdot}, i.e.,
how to actually use \eqref{eqn:pdot} to find the minimizer. We adopt a backtracking line search
algorithm with Armijo condition \cite{armijo1966minimization}. At time step $k$ with the current
approximation $p^k$, we introduce
\[
s^k = W \diag \left(
\frac{1}{\frac{\alpha_1}{H_1p^k} + \frac{\alpha_2}{H_2p^k} + \alpha_3 h_3}
\right)
W^\T \frac{\delta E}{\delta p}(p^k).
\]
Starting from $\eta=1$, one repetitively halves $\eta$ until
\[
E(p^k-\eta s^k) - E(p_k) \le -\frac{1}{2} \eta s^k\cdot \frac{\delta E}{\delta p}(p^k).
\]
Once it is reached, one sets
\[
p^{k+1}  = p^k - \eta s^k
\]
and move on to the next iteration until convergence.

\section{Numerical results} \label{sec:res}

This section presents several numerical examples to illustrate the efficiency of the combined
gradient descent \eqref{eqn:pdot} for the combined loss functionals.

\subsection{1D}
Consider first the 1D domain $\Omega=[0,1]$ with the periodic boundary condition.  Let $\mu$ be a
reference measure. Among the three terms of the combined loss functional $E(p)=\alpha_1
E_1(p)+\alpha_2 E_2(p)+\alpha_3 E_3(p)$, the first term $E_1(p)$ is a functional close to the square
of the 2nd Wasserstein distance $W_2(p,\mu)$ between $p$ and $\mu$. Because the exact computation of
$W_2^2(p,\mu)$ and its derivative with respect to $p$ is quite non-trivial, we replace $E_1(p)$ with
the square of the weighted semi $H^{-1}$-norm
\[
E_1(p) = \frac{1}{2} \norm{p-\mu}^2_{\dot{H}^{-1}(\mu)},
\]
which is known to be equivalent to the square of the $W_2$ norm \cite{peyre2018comparison}.
The $\dot{H}^1(\mu)$ for a signed measure $\eps$ is defined as
\[
\norm{\eps}_{\dot{H}^{-1}(\mu)} = \min_{\theta: \nabla\cdot(\mu \theta)=\eps} \int |\theta|^2 \d\mu,
\]
or equivalently
\[
\norm{\eps}_{\dot{H}^{-1}(\mu)} = \sup \left\{ (f,\eps) : \norm{f}_{\dot{H}^1(\mu)} \le 1 \right\},\quad
\norm{f}_{\dot{H}^1(\mu)} = \int |\nabla f|^2 \d\mu.
\]
After discretization, $E_1(p)$ takes the following simple form
\[
E_1(p) = \frac{1}{2} (p-\mu)^\T (D^\T \mu D)^+ (p-\mu).
\]
The second term $E_2(p)$ is the Kullback-Leibler divergence
\[
E_2(p) = \sum_s p_s \log \frac{p_s}{\mu_s}.
\]
Finally, the last term $E_3(p)$ is the Dirichlet energy given by
\[
E_3(p) =\frac{1}{2} (p-\mu)^\T (-\Delta) (p-\mu) = \frac{1}{2} (p-\mu)^\T D^\T D (p-\mu),
\]
so $A = (-\Delta)$. The minimizer of $E(p)$ is equal to $\mu$.

The domain is discretized with $n=512$ grid points. The reference measure $\mu(s) \sim \exp(-V(s))$
with $V(s) = \sin(4\pi s)$ for $s\in S$. The constant factors in front of the three terms are chosen
to be $1$, $10^{-3}$, and $10^{-4}$, respectively, in order to balance the contribution from three
terms so that none of them dominates. We test with four different linear combinations, with results
summarized in Figure \ref{fig:1D}.

\begin{figure}[h!]
  \centering
  \begin{tabular}{cc}
    \includegraphics[width=0.45\textwidth]{1D_1.pdf} &   \includegraphics[width=0.45\textwidth]{1D_2.pdf}\\
    (1) & (2)\\
    \includegraphics[width=0.45\textwidth]{1D_3.pdf} &  \includegraphics[width=0.45\textwidth]{1D_4.pdf}\\
    (3) & (4)
  \end{tabular}
  \caption{ (1) Wasserstein plus Fisher-Rao terms tested with Wasserstein GD, Fisher-Rao GD, and the
    combined natural gradient. (2) Wasserstein plus Mahalanobis terms tested with Wasserstein GD,
    Mahalanobis GD, and the combined natural gradient. (3) Fisher-Rao plus Mahalanobis terms tested
    with Fisher-Rao GD, Mahalanobis GD, and the combined natural gradient. (4) Wasserstein plus
    Fisher-Rao plus Mahalanobis terms tested with Wasserstein GD, Fisher-Rao GD, Mahalanobis GD, and
    the combined natural gradient.}
  \label{fig:1D}
\end{figure}

\begin{enumerate}
\item $(\alpha_1,\alpha_2,\alpha_3) = (1,10^{-3},0)$, i.e., turning off the Mahalanobis
  term. The combined natural gradient converges much more rapidly compared to the Wasserstein GD and
  the Fisher-Rao GD.
\item $(\alpha_1,\alpha_2,\alpha_3) = (1,0,10^{-4})$, i.e., turning off the Fisher-Rao term.
  The combined natural gradient converges much more rapidly compared to the Wasserstein GD and the
  Mahalanobis GD.
\item $(\alpha_1,\alpha_2,\alpha_3) = (0,10^{-3},10^{-4})$, i.e., turning off the Wasserstein
  term. The combined natural gradient converges much more rapidly compared to the
  Fisher-Rao GD and the Mahalanobis GD.
\item $(\alpha_1,\alpha_2,\alpha_3) = (1,10^{-3},10^{-4})$. The combined natural gradient
  converges much more rapidly compared to the Wasserstein GD, the Fisher-Rao GD, and the Mahalanobis
  GD.
\end{enumerate}

\subsection{2D}
Consider now the 2D domain $\Omega=[0,1]^2$ with periodic boundary condition.  Among the three terms
of the combined loss functional $E(p)=\alpha_1 E_1(p)+\alpha_2 E_2(p)+\alpha_3 E_3(p)$, $E_1(p)$ is
again chosen to be the weighted semi $H^{-1}$-norm
\[
E_1(p) = \frac{1}{2} \norm{p-\mu}_{\dot{H}^{-1}(\mu)}.
\]
After discretization, it takes the following form
\[
E_1(p) = \frac{1}{2} (p-\mu)^\T (D_1^\T \mu D_1 + D_2^\T \mu D_2)^+ (p-\mu)
\]
where $D_1$ and $D_2$ are the derivative operators in the first and the second directions.  $E_2(p)$
is again the Kullback-Leibler divergence
\[
E_2(p) = \sum_s p_s \log \frac{p_s}{\mu_s}.
\]
Finally, $E_3(p)$ is given by
\[
E_3(p) = \frac{1}{2} (p-\mu)^\T (-\Delta) (p-\mu) = \frac{1}{2} (p-\mu)^\T (D_1^\T D_1 + D_2^\T D_2) (p-\mu)
\]
so $A = (-\Delta)$.

The domain is discretized with $n=64$ grid point in each direction. $\mu(s_1,s_2)\sim
\exp(-V(s_1,s_2))$ with $V(s_1,s_2) = \sin(4\pi s_1)\sin(4\pi s_2)$. The constant factors of the
three terms are set to be $1$, $3\cdot10^{-4}$, and $10^{-4}$ in order to balance the contribution
from them so that no one dominates. We test with four different linear combinations, with the
results summarized in Figure \ref{fig:2D}.

\begin{figure}[h!]
  \centering
  \begin{tabular}{cc}
    \includegraphics[width=0.45\textwidth]{2D_1.pdf} &    \includegraphics[width=0.45\textwidth]{2D_2.pdf}\\
    (1) & (2)\\
    \includegraphics[width=0.45\textwidth]{2D_3.pdf} &    \includegraphics[width=0.45\textwidth]{2D_4.pdf}\\
    (3) & (4)
  \end{tabular}
  \caption{ (1) Wasserstein plus Fisher-Rao terms tested with Wasserstein GD, Fisher-Rao GD, and the
    combined natural gradient. (2) Wasserstein plus Mahalanobis terms tested with Wasserstein GD,
    Mahalanobis GD, and the combined natural gradient. (3) Fisher-Rao plus Mahalanobis terms tested
    with Fisher-Rao GD, Mahalanobis GD, and the combined natural gradient. (4) Wasserstein plus
    Fisher-Rao plus Mahalanobis terms tested with Wasserstein GD, Fisher-Rao GD, Mahalanobis GD, and
    the combined natural gradient.}
  \label{fig:2D}
\end{figure}

\begin{enumerate}
\item $(\alpha_1,\alpha_2,\alpha_3) = (1,3\cdot10^{-4},0)$, i.e., turning off the Mahalanobis
  term. The combined natural gradient converges much more rapidly compared to the Wasserstein GD and
  the Fisher-Rao GD.
\item $(\alpha_1,\alpha_2,\alpha_3) = (1,0,10^{-4})$, i.e., turning off the Fisher-Rao term. The
  combined natural gradient converges much more rapidly compared to the Wasserstein GD and the
  Mahalanobis GD.
\item $(\alpha_1,\alpha_2,\alpha_3) = (0,3\cdot10^{-4},10^{-4})$, i.e., turning off the
  Wasserstein term. The combined natural gradient converges much more rapidly compared to the
  Fisher-Rao GD and the Mahalanobis GD.
\item $(\alpha_1,\alpha_2,\alpha_3) = (1,3\cdot10^{-4},10^{-4})$. The combined natural gradient
  converges much more rapidly compared to the Wasserstein GD, the Fisher-Rao GD, and the Mahalanobis
  GD.
\end{enumerate}

\section{Discussions} \label{sec:disc}

This note proposes a new natural gradient for minimizing combined loss functionals by using
diagonal approximation in the wavelet basis. There are a few open questions. First, so far we have
considered regular domains in one and two dimensions with periodic boundary condition. One direction
is to extend this to more general domains using more sophisticated wavelet bases.

Second, we have assumed that the probability density $p$ is non-vanishing everywhere in deriving the
interpolating natural gradient metric. It is an important question whether one can remove this
condition in order to work with more general probability densities.

Third, the dynamics in the wavelet coefficients \eqref{eqn:cdot} enjoys a diagonal metric. It is
tempting to ask whether it is possible to design a mirror descent algorithm. Due to the coupling
between different wavelet coefficients in the metric computation $H_1 Wc$ and $H_2 Wc$, this seems
quite difficult. An interesting observation is that the metric of the coarse scale wavelet
coefficients is nearly independent of the values of the fine scale coefficients, while the metric of
the fine scale ones depends heavily on the values of the coarse scale ones. This naturally brings
the question of whether the combined metric (or even the Wasserstein metric) has an approximate
multiscale structure. The wavelet analysis has played an important role understanding the earth
mover distance metric $W_1$ \cite{indyk2003fast,shirdhonkar2008approximate}. It seems that it might
also play a role in understanding the $W_2$ metric.

\bibliographystyle{abbrv}

\bibliography{ref}

\end{document}

config: arxiv


text:
\begin{document}
\pagestyle{plain}
\title{\vskip -70pt
\begin{flushright}
{\normalsize DAMTP-2010-50}\\
\end{flushright}
\vskip 8pt
{\bf Cosmological Einstein--Maxwell Instantons and Euclidean
Supersymmetry:\\ Anti--Self--Dual Solutions.}\vskip 3pt}
\author{Maciej\ Dunajski
\thanks{email {\tt m.dunajski@damtp.cam.ac.uk}}\\
{\sl Department of Applied Mathematics and Theoretical Physics} \\
{\sl University of Cambridge} \\
{\sl Wilberforce Road, Cambridge CB3 0WA, UK.} \\
{and}\\
{\sl Institute of Mathematics of the  Polish Accademy of Sciences},\\
{\sl \'Sniadeckich 8, 00-956 Warsaw, Poland}.
\\[8pt]
Jan \ Gutowski \thanks{email {\tt jan.gutowski@kcl.ac.uk}}\\
{\sl
Department of Mathematics, King' s College London }\\
{\sl Strand, London WC2R 2LS, UK.}\\[8pt]
Wafic \ Sabra \thanks{email {\tt ws00@aub.edu.lb}}
\\[3pt]
{\sl  Centre for Advanced Mathematical Sciences and Physics Department}\\[1pt]
{\sl American University of Beirut, Beirut, Lebanon.}\\[8pt]
Paul \ Tod \thanks{email {\tt paul.tod@sjc.ox.ac.uk}}\\
{\sl The Mathematical Institute, Oxford University  }\\
{\sl 24-29 St Giles, Oxford OX1 3LB, UK.}}
\date{}
\maketitle
\thispagestyle{empty}
\begin{abstract}
We classify super--symmetric solutions of the minimal $N=2$ gauged
Euclidean supergravity in four dimensions. The solutions
with anti--self--dual Maxwell field give rise to anti--self--dual Einstein
metrics given in terms of solutions to the
$SU(\infty)$ Toda equation and more general three--dimensional Einstein--Weyl structures. Euclidean Kastor--Traschen metrics
are also characterised by the existence of a certain super covariantly constant spinor.
\end{abstract}
\newpage

\section{Introduction}

The bosonic sector of $N=2$ supergravity (SUGRA) in four dimensions
coincides with the Einstein--Maxwell theory. In \cite{Tod83} all solutions
which admit a supercovariantly
constant  spinor have been found.

In this work we  shall classify supersymmetric
solutions of  Euclidean Einstein--Maxwell equations with
non--zero cosmological constant. It will be shown that the field
equations in various branches of our classification reduce to the
Einstein--Weyl system in three dimensions \cite{Hi,JT85,Dbook}
which is integrable by twistor construction. Some of the Euclidean solutions
arise from analytic continuations of real Lorentzian solutions - for example
the Euclidean analogs of Kastor--Traschen metrics \cite{KS}
belong to this class - while others do not have Lorentzian counterparts.
In particular all solutions with anti--self--dual Maxwell field belong to the
latter class. It turns out (Proposition \ref{prop11})
that the anti--self--duality of the Maxwell
field implies the conformal anti--self--duality (ASD) of the Weyl tensor. In this
paper we shall focus on constructing all solutions belonging to this ASD class.
The non anti--self--dual solutions will be constructed in \cite{DGST2}.
Some of these have a Lorentzian counterpart \cite{CK, CK1, other_paper, GS, Grov1,
Grov2}.

\vskip5pt
In the  second part of this introduction we
shall discuss the Euclidean Einstein--Maxwell theory and explain the
origin of various sign choices in the Euclidean signature.
In Section 2  we shall use the two--component spinor calculus to classify all
supersymmetric solutions. The Killing spinor equations  (\ref{spinor_cond})
contain a continuous
parameter and we shall show that the Killing spinor gives rise to
a Killing vector only for one special value of this parameter.
In this symmetric case the metric is given in terms of solutions to the
$SU(\infty)$ Toda equation (Proposition \ref{prop_toda}).
For all other values of the parameter the solutions do not in general
admit an isometry.
They do however admit a conformal retraction (Proposition \ref{propGT1}
and Proposition \ref{prop_interpol}).
In Section 3 we shall characterise the Euclidean Kastor--Traschen solutions
by the existence of a supercovariantly constant spinor with certain additional
properties (Proposition \ref{propKT}). The solutions constructed in this
section are not anti--self--dual.

\vskip5pt
There are several  motivations for studying Euclidean
gauged SUGRA solutions.
From the differential geometric perspective the supersymmetric
solutions constructed in Proposition \ref{propGT1} and Proposition
\ref{prop_interpol} provide
examples of anti--self--dual Einstein metrics. The point is that
the energy--momentum tensor of the ASD Maxwell field vanishes and the Maxwell
equations decouple from the Einstein equations.
In Euclidean Quantum gravity instantons provide
semi--classical description of black hole creations and in the cosmological
context  this has been implemented  in \cite{HR, Romans, Ross-Man, MM89}.
Finally solutions of  ungauged ($\Lambda=0$) SUGRA can be used to construct
supersymmetric solutions of Lorentzian minimal SUGRA theories in five
and higher dimensions \cite{DH07, Bobev1, Bobev2}. It remains to be seen
whether solutions to the gauged $D=4$ Euclidean SUGRA admit such lifts.
\newpage
{\bf Acknowledgements.} MD, JG and PT
thank the American University of Beirut
for hospitality when some of this work was carried over.
The work of WS is supported in part by the National Science
Foundation under grant number PHY-0903134. JG is supported by EPSRC grant EP/F069774/1. The authors thank David Calderbank for the interesting comments on the manuscript.
\subsection{Euclidean Einstein--Maxwell equations}
Consider Lorentzian Einstein--Maxwell equations possibly with non-vanishing cosmological constant
\be
\label{lorentzian_EM}
G_{ab}+6\Lambda g_{ab}=-T_{ab}, \qquad d F=0, \qquad d*F=0,
\ee
where
\[
T_{ab}= \frac{1}{2}g_{ab}|F|^2-2F_{ac}{F_b}^c
\]
is the Maxwell energy--momentum tensor\footnote{
Our conventions follow Penrose and Rindler \cite{PR}: $[\nabla_a, \nabla_b]V^d={R_{abc}}^dV^c,
R_{ab}={R_{acb}}^c=6\Lambda g_{ab}-2\Phi_{ab}$, where $\Phi_{ab}$ is the traceless Ricci tensor. Using these conventions in the Riemannian settings implies that the hyperbolic space
$H^4$ has $\Lambda>0$ and the four-sphere $S^4$ has $\Lambda<0$.} and
$|F|^2=F_{cd}F^{cd}$. Swapping the electric and magnetic fields, i. e.
replacing $F$ by
its Hodge dual $*F$ maps solutions to solutions as the Lorentzian Maxwell E-M tensor is unchanged by this transformation. This can be easily seen in the
two--component spinor notation \cite{PR} where
\[
T_{ab}=2\phi_{AB}\overline\phi_{A'B'}
\]
and
the duality transformation is $\ov{\phi}_{A'B'}\rightarrow i\ov{\phi}_{A'B'}$
and
$\phi_{AB}\rightarrow -i\phi_{AB}$.
This is no longer the case in the Riemannian signature where
\[
T_{ab}=2\phi_{AB}\tilde\phi_{A'B'}
\]
and the spinors $\phi_{AB}$ and $\tilde{\phi}_{A'B'}$ are independent. The transformation
$F\rightarrow *F$ entails to $\tilde{\phi}_{A'B'}\rightarrow \tilde{\phi}_{A'B'}$
and ${\phi}_{AB}\rightarrow -{\phi}_{AB}$, thus $T_{ab}\rightarrow -T_{ab}$.
This duality transformation can be used to `fix wrong signs' arising from the
analytic continuation of a Lorentzian solution.

As an example
consider the Reissner--Nordstr\"om--de Sitter
space--time
\be
\label{RSDS}
g=-V(r)dt^2+V(r)^{-1}dr^2+r^2(d\theta^2+\sin^2{\theta} d\phi^2),\qquad A=-Q\frac{dt}{r},
\ee
where
\[
V(r)=1-\frac{2m}{r}+\frac{Q^2}{r^2}-2\Lambda r^2,
\]
$m\geq 0$ and $Q$ are constants
and $F=dA$. Now continue this analytically to the Riemannian signature setting $t=i\tau$
and  assuming that $r$ lies between the middle roots of
the quartic $r^2V(r)=0$.
We encounter an immediate problem as the potential $A$ is now purely imaginary.
There does not seem to be a universally accepted resolution of this problem, and the way in which one proceeds is dictated by an overall aim of the analytic continuation.
According to Hawking and Ross \cite{HR} one should, at least in the Quantum--Mechanical context, accept that the electrically charged solution has imaginary gauge potential.
Alternatively, especially if our interest lies in classical solutions, one could
replace $A$ by $iA$ which is real. However this continuation changes the overall
sign of $T_{ab}$ and leads to a `wrong' coupling between the gravitational and electromagnetic fields. The coupling can now be `made right' by replacing
$F\rightarrow *F$, resulting in the Maxwell field $F=-*d(Qr^{-1}d\tau)$.

In this discussion we used `wrong' and `right' in inverted commas, as the energy of the Riemannian Maxwell field is not positive definite, and (unlike in the Lorentzian case) the positivity can not be used to fix the relative sign between $G_{ab}$ and  $T_{ab}$.
The cosmological constant in our example doesn't change under the
analytic continuation.
In Section \ref{sectionthree} we shall however see a different class of examples
(Kastor--Traschen cosmological multi black holes \cite{KS}) where the analytic continuation leads to a real solution only if $\Lambda$ changes sign: asymptotically de Sitter Lorentizan
solutions become asymptotically hyperbolic Riemannian solutions.

To avoid making the various sign choices we shall simply look for real solutions of the Euler--Lagrange equations arising from the Lagrangian density
\[
{\cal L}=\sqrt{g}(R+\gamma |F|^2+\delta),
\]
where $\gamma$ and $\delta$ are real constants. The cosmological constant can then be read--off from $\delta$ and the sign of the Maxwell energy--momentum tensor can be adjusted if necessary replacing $F$ by its Hodge dual as explained above.

\section{Supersymmetric solutions with ASD Maxwell field}
Let the two--form $F$ be an anti--self--dual (ASD) Maxwell field on a
Riemannian four--manifold $(M, g)$, i. e.
\[
d F=0, \qquad *F=-F.
\]
We shall make use of an isomorphism
\[
\C\otimes T M\cong {\spp}\otimes {\spp'}
\]
where the  complex rank--two vector bundles
$\spp, \spp'$  (called spin-bundles) over $M$
are equipped with parallel symplectic
structures $\ep, \ep'$ such that $g=\vv\otimes \vv'$.
We use the standard convention \cite{PR, Dbook} in which spinor indices are capital
letters, unprimed for sections of $\spp$ and primed for sections of $\spp'$.
The two component spinor formalism will be adapted to the Riemannian signature, where the spinor conjugation
preserves the type of spinors. Thus if $\alpha_{A}=(p, q)$ we can
define $\hat{\alpha}_A=(\overline{q}, -\overline{p})$ so that
${\hat{\hat{\alpha}}}_A=-\alpha_A$. This Hermitian conjugation
induces a positive inner product
\[
\hat{\alpha}_A{\alpha}^A=\ep_{AB}\hat{\alpha}^A{\alpha}^B=|p|^2+|q|^2 \,.
\]
We define the inner product on the primed spinors in the same way.
Here $\vv_{AB}$ and $\vv_{A'B'}$  are covariantly constant symplectic
forms with $\vv_{01}=\vv_{0'1'}=1$. These are used to raise and lower spinor indices
according to $\alpha_B=\ep_{AB}\alpha^A, \alpha^B=\ep^{BA}\alpha_A$, and
similarly for primed spinors.

The spinor decomposition
of the Riemann tensor is
\begin{eqnarray} \label{riemann}
R_{abcd} &=& \psi_{ABCD} \ep_{A'B'} \ep_{C'D'} + {\psi}_{A'B'C'D'}
\ep_{AB} \ep_{CD} \nonumber\\
&&+ \Phi_{ABC'D'} \ep_{A'B'}\ep_{CD} +
\Phi_{A'B'CD} \ep_{AB}\ep_{C'D'} \nonumber\\&& + 2\Lambda (\ep_{AC}\ep_{BD}
\ep_{A'B'} \ep_{C'D'} - \ep_{AB} \ep_{CD} \ep_{A'D'} \ep_{B'C'}),
\end{eqnarray}
where $\psi_{ABCD}$ and $\psi_{A'B'C'D'}$ are ASD and SD Weyl spinors
which are symmetric in their indices,
$\Phi_{A'B'CD}=\Phi_{(A'B')(CD)}$ is the traceless Ricci spinor and $\Lambda=R/24$ is the cosmological constant.

Making use of the isomorphism $\Lambda^2_-\cong \spp\odot \spp$
we can write $F_{ab}=\phi_{AB}\varepsilon_{A'B'}$ where the symmetric spinor
$\phi_{AB}=\hat{\phi}_{AB}$ satisfies the ASD Maxwell equations
\[
{\nabla^A}_{A'} \phi_{AB}=0.
\]

Consider the Killing spinor equations \cite{GHull,Tod83}
\begin{eqnarray}
\label{spinor_cond0}
\nabla_{AA'}\alpha_B+c_0 A_a\alpha_B+(c_1\phi_{AB}+c_2\vv_{AB})\beta_{A'}&=&0,\nonumber\\
\nabla_{AA'}\beta_{B'}+c_3 A_a\beta_{B'}+c_4\vv_{A'B'}\alpha_A&=&0,
\end{eqnarray}
where $A_a$ is a real one--form and
$c_0, \dots, c_4$ are some constant coefficients which we shall now
determine.  Differentiating (\ref{spinor_cond0}) covariantly,
commuting covariant derivatives and using the
spinor Ricci identities
\be
\label{cc1}
{\nabla^A}_{(A'}\nabla_{B')A}\alpha_B+\Phi_{A'B'AB}\alpha^A=0,
\ee
\be
\label{cc2}
{\nabla^{A'}}_{(A}\nabla_{B)A'}\beta_{B'}+\Phi_{A'B'AB}\beta^{A'}=0,
\ee
\be
\label{cc3}
{\nabla^{A'}}_{(A}\nabla_{B)A'}\alpha_{C}+\psi_{ABCD}\alpha^D- 2\Lambda\alpha_{(A}\vv_{B)C}=0,
\ee
\be
\label{cc4}
{\nabla^{A}}_{(A'}\nabla_{B')A}\beta_{C'}+\psi_{A'B'C'D'}\beta^{D'}- 2\Lambda\beta_{(A'}\vv_{B')C'}=0,
\ee
leads
to the  compatibility conditions:
Equations ({\ref{cc1}}) give
\[
\Phi_{ABA'B'}=0, \qquad c_0{\nabla^A}_{(A'}A_{B')A}=0, \qquad c_0=c_3.
\]
Equations ({\ref{cc2}}) give $F=2dA$ if
${\nabla^A}_{(A'}A_{B')A}=-\tilde{\phi}_{A'B'},
{\nabla^{A'}}_{(A}A_{B)A'}=-{\phi}_{AB}
$ or
\[
c_3=-c_1c_4.
\]
Equations ({\ref{cc3}}) give
\[
c_2c_4=-\Lambda.
\]
Finally equations ({\ref{cc4}}) give
\[
\psi_{A'B'C'D'}=0
\]
so we deduce
\begin{prop}
\label{prop11}
A Riemannian four--manifold which admits a solution to
the Killing spinor equations
{\em(\ref{spinor_cond0})} with anti--self--dual Maxwell field $F$
is anti--self--dual and Einstein.
\end{prop}
\vskip10pt
The case $c_0=c_1=c_3=0$ leads to
some non--trivial
solutions in $(2, 2)$ signature, but not in $(4, 0)$ so we shall not
consider it.
If $c_0\neq 0$, then we can redefine $A_a$, $\phi_{AB}$ and $\beta_{A'}$ by rescalings to get rid
of some constants. Set $c_0=-ce^{i\theta}$, where $c$ and $\theta$ are real. The resulting equations are
\begin{eqnarray}
\label{spinor_cond}
\nabla_{AA'}\alpha_B&=&e^{i\theta}A_a\alpha_B+\Big(\frac{e^{i\theta}}{\Lambda}
\phi_{AB}-\vv_{AB}\Big)\beta_{A'}
\\
\nabla_{AA'}\beta_{B'}&=&e^{i\theta} A_a\beta_{B'}+\Lambda\vv_{A'B'}\alpha_A, \nonumber
\end{eqnarray}
together with equations for spinor conjugates
\begin{eqnarray}
\label{spinor_cond1}
\nabla_{AA'}\hat{\alpha}_B&=&e^{-i\theta}A_a\hat{\alpha}_B+\Big(\frac{e^{-i\theta}}{\Lambda}\phi_{AB}-
\vv_{AB}\Big)\hat{\beta}_{A'}
\\
\nabla_{AA'}\hat{\beta}_{B'}&=&e^{-i\theta} A_a\hat{\beta}_{B'}+\Lambda\vv_{A'B'}\hat{\alpha}_A.  \nonumber
\end{eqnarray}
\subsection{$\theta=\pi/2$ and $SU(\infty)$ Toda equation}
Now we shall consider the case $\theta=\pi/2$ and show that the resulting metric is the most general ASD Einstein metric with symmetry, and can be found from solutions to $SU(\infty)$ Toda equation.
\begin{prop}
\label{prop_toda}
Let the Riemannian four--manifold $(M, g)$ admit a solution to the
Killing spinor equations {\em(\ref{spinor_cond})} with $\theta=\pi/2$ such that
$F_{ab}=\phi_{AB}\vv_{A'B'}$ is an ASD Maxwell field with $F=2dA$. Then $g$ satisfies
ASD Einstein equations with non--zero $\Lambda$. Moreover $g$ admits a Killing vector
and local coordinates $(x, y, z, \tau)$ can be chosen so that
\be
\label{prop1_metric}
g=\frac{1}{z^2}\Big(V(dz^2+e^u(dx^2+dy^2))+V^{-1}(d\tau+\omega)^2\Big),
\ee
where $u=u(x, y, z)$  is a solution of the $SU(\infty)$ Toda equation
\be
\label{todaeq}
u_{xx}+u_{yy}+({e^u})_{zz}=0,
\ee
the function $V$ is given by $-4\Lambda V=zu_z-2$, and $\omega$ is a
one--form such that
\be
\label{domega}
d\omega=-V_xdy\wedge dz-V_y dz\wedge dx -(Ve^u)_zdx\wedge dy.
\ee
\end{prop}
We have already shown that $g$ is ASD and Einstein. Once we establish
the existence of a symmetry, we could refer to results of Tod \cite{T97} and Przanowski
\cite{Prz} to deduce the canonical form of the metric (\ref{prop1_metric}).
In the proof below we shall however give
a direct derivation of this form using the Killing spinor equations.

{\bf Proof.} Define two real non--zero functions $U, \widetilde{U}$ by
\be
\label{UUbar}
U=(\varepsilon_{AB}\hat{\alpha}^A\alpha^B)^{-1}, \qquad \widetilde{U}=(\varepsilon_{A'B'}\hat{\beta}^{A'}\beta^{B'})^{-1}.
\ee
Consider a complex tetrad of one--forms
\be
\label{tetrad_1}
K_a=i(\hat{\alpha}_A\beta_{A'}+{\alpha}_A\hat{\beta}_{A'}),
\quad X_a=\hat{\alpha}_A\beta_{A'}-{\alpha}_A\hat{\beta}_{A'}, \quad
Z_a=\alpha_A\beta_{A'}.
\ee
The one--forms $X=X_ae^a, K=K_ae^a$ are real and the one--form
$Z=Z_ae^a$ is complex. Using the Killing spinor equations (\ref{spinor_cond}) and their conjugations (\ref{spinor_cond1})
we find
\be
\label{dK}
\nabla_aK_b=\vv_{A'B'}(\Lambda(\hat{\alpha}_A\alpha_B+\alpha_A\hat{\alpha}_B)-\frac{i}{\Lambda\widetilde{U}}\phi_{AB})-\varepsilon_{AB}(\hat{\beta}_{A'}\beta_{B'}+\beta_{A'}\hat{\beta}_{B'})
\ee
so that $\nabla_{(a} K_{b)}=0$ and $K$ is a Killing vector. Moreover we find
\[
dX=0, \qquad Z\wedge dZ=0
\]
and deduce existence of a local coordinate system $(\tau, \zeta, q, \ov{q})$ on $M$ such that
\[
K^a\nabla_a=\sqrt{2}\frac{\p}{\p\tau}, \quad X=\sqrt{2}d\zeta, \quad  Z=\frac{1}{\sqrt{2}}pdq
\]
for some complex--valued function $p=p(\zeta, q, \ov{q})$.
Therefore the one--form dual to the Killing vector is $K=\Omega(d\tau+\omega)$,
where $\Omega$ and $\omega$ are a function and a one--form on the space of orbits
of $K$ in $M$. Using
\be
\label{epsilons}
\varepsilon_{AB}=U(\hat{\alpha}_A\alpha_B-\hat{\alpha}_B\alpha_A),
\quad
\varepsilon_{A'B'}=\widetilde{U}(\hat{\beta}_{A'}\beta_{B'}
-\hat{\beta}_{B'}\beta_{A'})
\ee
we find the metric to be
\[
g=\vv_{AB}\vv_{A'B'}e^{AA'}e^{BB'}=U\widetilde{U}(d\zeta^2+|p|^2dqd\ov{q}+\Omega^2(d\tau+\omega)^2).
\]
Finally using $K_aK^a=2(U\widetilde U)^{-1}$ and setting $q=x+iy$,
$|p|^2=e^{\phi}$ where
$\phi=\phi(x, y, \zeta)$ is a real--valued function yields
\[
g=U\widetilde{U} (e^\phi(dx^2+dy^2)+d\zeta^2)+\frac{1}{U\widetilde{U}}(d\tau+\omega)^2.
\]
Now we need to find equations for $\phi, U, \widetilde{U}$ and $\omega$.
Using the Killing spinor equations (\ref{spinor_cond}) gives
\be
\label{UtildeU}
\nabla_a\Big(\frac{1}{\widetilde{U}}\Big)=\Lambda X_a, \quad
\nabla_a\Big(\frac{1}{U}\Big)=-\frac{1}{\Lambda}{\phi_A}^CK_{CA'}+X_a.
\ee
Therefore $\widetilde{U}=(\sqrt{2}\Lambda\zeta)^{-1}$, where we absorbed the integration constant
into the definition of $\zeta$. Defining a coordinate $z=\zeta^{-1}$ and setting
\[
U=\sqrt{2}\Lambda z V, \quad \phi=u+4\log{\zeta}
\]
where $V=V(x, y, z), u=u(x, y, z)$
yields the final form of the metric (\ref{prop1_metric}).
We now use $(\ref{UtildeU})$ to find
\[
\phi_{AB}=\frac{2\Lambda}{|K|^2}K^{A'}_B\nabla_{AA'}
\Big(\frac{1}{U}-\frac{\Lambda}{\widetilde{U}}\Big),
\]
and substitute this to (\ref{dK}). This yields $-4\Lambda V=zu_z-2$,
where $u$ satisfies
the SU$(\infty)$ Toda equation (\ref{todaeq}), and (\ref{domega}) holds.
\koniec
Note that the rescaled metric $\hat{g}=z^2g$ is of the form given by the LeBrun ansatz \cite{lebrun}
because $V$ satisfies the linearised
$SU(\infty)$ Toda equation. Therefore
$\hat{g}$ is K\"ahler with vanishing Ricci scalar.
Scalar--flat K\"ahler metrics are also solutions to Einstein--Maxwell equations
in the Riemannian signature \cite{flaherty}, where the self--dual (SD)
and anti--self--dual (ASD) parts of
the Maxwell field are given  by the K\"ahler form $\Omega$ and (half of)
the Ricci form $\rho$ respectively
\[
F=\Omega+\frac{\rho}{2}.
\]
Note that $\Omega\in \Lambda^2_+$ and $\rho\in\Lambda^2_-$.
Thus there exist two conformally related metrics: one non-supersymmetric $\hat{g}$ which solves Euclidean ungauged supergravity equations (Einstein--Maxwell
with $\Lambda=0$),
and one supersymmetric $g$ which solves
the gauged supergravity (Einstein--Maxwell
with $\Lambda\neq0$).
\vskip10pt

\vskip10pt
\subsection{$\theta=0$ and the hyperCR equation}
The ASD Einstein metrics corresponding to $\theta\neq \pi/2$ in (\ref{spinor_cond}) do not in general
admit a continuous symmetry. In this subsection we shall find a general local form of the metric in the case
when $\theta=0$.
\begin{prop}
\label{propGT1}
Let the Riemannian four--manifold $(M, g)$ admit a solution to the
Killing spinor equations {\em(\ref{spinor_cond})} with $\theta=0$ such that
$F_{ab}=\phi_{AB}\vv_{A'B'}$ is an ASD Maxwell field with $F=2dA$. Then $g$ satisfies
ASD Einstein equations with $\Lambda>0$. Moreover a local coordinate $\psi$ can be chosen so that
\be
\label{GTtheta0}
g=\frac{\Lambda}{8}\sinh{(2\psi)}^2h+\frac{2}{\Lambda}(d\psi-\coth{(\psi)}\omega)^2, \qquad F=2\;d(\coth{(\psi)}^2\omega)
\ee
where $h=\e_1^2+\e_2^2+\e_3^2$ is a 3-metric, the $\psi$--independent one--forms $(\e_i, \omega)$ satisfy
$\p_\psi\hook \e_i=\p_\psi\hook\omega=0$,
\begin{eqnarray}
\label{tod_eq}
d\e_1&=&-2\omega\wedge \e_1-\Lambda \e_2\wedge \e_3,\nonumber\\
d\e_2&=&-2\omega\wedge \e_2-\Lambda \e_3\wedge \e_1,\\
d\e_3&=&-2\omega\wedge \e_3-\Lambda \e_1\wedge \e_2\nonumber,
\end{eqnarray}
and
\be
\label{om_eq1}
d\omega=\Lambda *_h\omega,
\ee
where $*_h$ is the Hodge operator of $h$.
\end{prop}
{\bf Proof.}
The ASD Einstein equations follow from the integrability conditions as we have already explained.

The gauge freedom
\[
\alpha_A\rightarrow e^f\alpha_A, \quad \beta_{A'}\rightarrow e^f\beta_{A'}, \quad
A\rightarrow A-df, \qquad\mbox{where}\quad f:M\longrightarrow \R
\]
can be used to set $\widetilde{U}=1$. Consider a tetrad (\ref{tetrad_1}), so that
with our gauge choice
\[
g_{ab}=\frac{U}{2}(4Z_{(a}\ov{Z}_{b)}+K_aK_b+X_aX_b)
\]
and $X_aX^a=K_aK^a=2Z_a\overline{Z}^a=2U^{-1}$ and all other inner product vanish.

The condition $d(\widetilde{U}^{-1})=0$ implies
\[
A_a=\frac{\Lambda}{2}X_a.
\]
We also find
\[
X^a\nabla_a (U^{-1})=2U^{-1}(\Lambda U^{-1}-1),
\]
so that if $\tau$ is a local coordinate for which $X^a\nabla_a=\p/\p\tau$ then
\be
\label{formula_for_U}
U=\Lambda(1+e^{2\tau}c^2),
\ee
where $c$ is a local function on $M$ independent on $\tau$. Now we use the Killing spinor equations (\ref{spinor_cond}) and (\ref{epsilons}) to find
\begin{eqnarray}
\label{dzdx}
dZ&=&(-UX+i(U-\Lambda)K)\wedge Z\nonumber\\
dK&=&-UX\wedge K+2i(U-\Lambda) Z\wedge\overline{Z},
\end{eqnarray}
so regarding $X=\p/\p \tau$ as a vector field
\[
{\cal L}_XZ=-2Z, \qquad {\cal L}_X K=-2K
\]
where ${\cal L}_X$ denotes the Lie derivative along the vector field $X=X^a\nabla_a$.
Therefore we can set
\[
Z=e^{-2\tau}\widetilde{Z},\quad K=e^{-2\tau}\widetilde{K}
\]
where $\widetilde{Z}$ and $\widetilde{K}$ are one--forms which Lie--derive along $X$.
The one form $X_a$ is given by $X=2U^{-1}(d\tau+\Omega)$, where $\Omega$ is a one--form
which in general can depend on $\tau$.
We now have to consider two cases
\begin{enumerate}
\item $U=\Lambda$, which corresponds to  vanishing of the function $c$ in (\ref{formula_for_U}). Now
\[
d\widetilde{Z}=-2\Omega\wedge\widetilde{Z}, \quad d\widetilde{K}=-2\Omega\wedge\widetilde{K},
\]
so that $\Omega$ is independent on $\tau$. Taking the exterior derivatives of these
equations gives the integrability condition $d\Omega=0$. Therefore locally there exist
 $\tau$--independent real valued functions $(\phi, x, y, z)$
such that
\[
\Omega=d\phi, \quad \widetilde{Z}= \frac{1}{2}e^{-2\phi}(dx+id y), \quad
K=e^{-2\phi} d z.
\]
Finally setting $\tilde{\tau}=\tau+\phi$ gives the
hyperbolic metric
\be
\label{hyperbolic_g}
g=\frac{\Lambda}{2}e^{-4\tilde{\tau}}(dx^2+dy^2+dz^2)+\frac{2}{\Lambda}d\tilde{\tau}^2
\ee
and the vanishing Maxwell field $F=0$. This metric  has $\Lambda>0$ which is consistent with our curvature conventions.
\item Now assume $U\neq\Lambda$. Equations (\ref{dzdx}) imply
\begin{eqnarray}
\label{neweq}
d\widetilde{Z}&=&(-2\Omega+i\Lambda c^2\widetilde{K})\wedge\widetilde{Z}\nonumber\\
d\widetilde{K}&=&-2\Omega\wedge\widetilde{K}+2i\Lambda c^2\widetilde{Z}\wedge\overline{\widetilde{Z}}.
\end{eqnarray}
We can redefine coordinates to set $c=1$. To see it put
\be
\label{new_eqqq}
\widetilde{Z}=\frac{c^{-2}}{2}(\e_1+i\e_2), \quad \widetilde{K}=c^{-2}\e_3, \quad
\tilde{\tau}=\tau-\log{c}, \quad {\omega}=\Omega+d\log{c}.
\ee
Then the metric is given by
\[
g=\Lambda(1+e^{2\tilde{\tau}})\Big(\frac{1}{2}e^{-4\tilde{\tau}}{h}+\frac{2}{\Lambda^2(1+e^{2\tilde{\tau}})^2}(d\tilde{\tau}+{\omega})^2\Big),
\]
where $h=\e_1^2+\e_2^2+\e_3^2$.
Substituting (\ref{new_eqqq}) into (\ref{neweq}) gives the system
(\ref{tod_eq}) for the one--forms $\e_i$ .The Maxwell
field is given by
\[
F=d\Big(\frac{2\omega}{1+e^{2\tilde{\tau}}}\Big)
\]
and the anti--self--duality condition $F=-*F$ yields (\ref{om_eq1}).
This is also the integrability condition for (\ref{tod_eq}).
Setting $\psi=-\mbox{arctanh}(\sqrt{1+e^{2\tilde{\tau}}}) $ yields
the form of the metric and the Maxwell field given in the statement of the
Proposition.
\end{enumerate}
\koniec
{\bf Remarks}
\begin{itemize}
\item
Making an analytic continuation $\psi\rightarrow i\psi$ and
changing the sign of $\Lambda$ leads to an ASD Einstein metric given in terms
of trigonometric (rather than hyperbolic) functions. Setting $\Lambda=-4$
yields
\[
g=\frac{1}{4}\sin^2{(2\psi)}h+\frac{1}{4}(d\psi+\cot{\psi}\;\omega)^2
\]
\item A three--manifold admitting a system of one--forms $(\e_i, \omega)$
satisfying equations (\ref{tod_eq}) and (\ref{om_eq1}) admits a hyper--CR Einstein--Weyl
(EW) structure \cite{GT98}. There is a well--known construction \cite{JT85} which associates
ASD conformal structures with symmetry to any EW structure. Proposition \ref{propGT1}
reveals another connection between the hyperCR EW structures and ASD four--manifolds,
where the Einstein metric in an ASD conformal class does not admit a symmetry.
\item In \cite{DT99} it was shown how to reduce the hyperCR conditions
(\ref{tod_eq}) and (\ref{om_eq1})
to a single second--order
integrable PDE (which therefore plays a role analogous to the $SU(\infty)$ Toda equation) for one function of three variables.
\end{itemize}
The metric (\ref{GTtheta0}) degenerates at $\psi=0$ but this degeneracy can be absorbed into a conformal factor as
\[
g=\sinh{(2\psi)}^2 \hat{g},\quad \mbox{where}\quad \hat{g}=\frac{\Lambda}{8}h+\frac{2}{\Lambda}(d\chi+e^{-2\chi}\sinh{(2\chi)}\;\omega)^2
\]
and $\chi=-\mbox{arctanh}{(e^{2\psi})}$. The conformal structure will therefore be regular if
the pair $(h, \omega)$ is. An example is provided by the Berger sphere,
where\footnote{The equations (\ref{tod_eq}) hold, but not in the `obvious' frame
$\e_1=\sigma_1, \e_2=\sigma_2, \e_3=\Lambda\sigma_3$.
See \cite{Chave} for relevant formulae.}
\[
h=(\sigma_1)^2+(\sigma_2)^2+ \Lambda^2(\sigma_3)^2, \qquad
\omega=\frac{1}{2}\Lambda\sqrt{1-\Lambda^2}\sigma_3,
\]
where $0<\Lambda\leq 1$ and $\sigma_i$ are the left--invariant one--forms on
$S^3$ satisfying
\[
d\sigma_1=\sigma_2\wedge\sigma_3, \quad
d\sigma_2=\sigma_3\wedge\sigma_1, \quad
d\sigma_3=\sigma_1\wedge\sigma_2.
\]

\subsection{General $\theta$ and interpolating Einstein--Weyl equations}
Finally we shall analyse the Killing spinor equations
(\ref{spinor_cond}) and  (\ref{spinor_cond1}), where the parameter
$\theta$ is allowed to take  arbitrary values. Similarly  to the cases $\theta=\pi/2$ and
$\theta=0$ we shall find that the space--time admits a local fibration over a three--dimensional manifold with an Einstein--Weyl structure. The relevant Einstein--Weyl
structure has arisen in \cite{DT99} as the most general symmetry reductions of ASD Ricci--flat equations by a conformal Killing vector.  It contains both the $SU(\infty)$ and hyperCR equations as special cases. The class of
ASD Einstein metrics characterised in the following Proposition
does not in general
admit an isometry (or a conformal isometry) unless $\theta=\pi/2$. Instead
it will be shown to admit an ASD conformal retraction in a sense of \cite{Hi} and
\cite{Cald} (in \cite{Cald} it is
referred to  a conformal submersion. The metrics
from Proposition \ref{propGT1}
 belong to the class described in Theorem IX in this reference. The metrics characterised by the proposition below appear to be new).
\begin{prop}
\label{prop_interpol}
Let $(M, g)$ be a Riemannian four--manifold which admits a solution
of the Killing spinor equations  {\em(\ref{spinor_cond})} and  {\em(\ref{spinor_cond1})}
such that the two--form $F=2dA$ is anti--self--dual. Then $g$ is
anti--self--dual and Einstein with $\Lambda\neq 0$ and locally
is of the form
\be
\label{last_metric}
g=
\frac{2}{\Lambda}\Big(
\frac{e^{2\cos{\theta}\tau}}{1+e^{2\cos{\theta}\tau}}\Big)
\Big(d\tau-\omega+\frac{1}{2}\Lambda \tan{\theta}\; e^{2\cos{\theta}\tau}
\; \e_3\Big)^2+ \frac{\Lambda}{2} e^{4\cos{\theta}\tau}(1+e^{-2\cos{\theta}\tau})h
\ee
where $h=\e_1^2+\e_2^2+\e_3^2$ is a 3-metric, the $\tau$--
independent one--forms $(\e_i, \omega)$ satisfy
$\p_\tau\hook \e_i=\p_\tau\hook\omega=0$, and
\begin{eqnarray}
\label{ewequations}
d\e_3&=&-2\cos{\theta} \omega\wedge \e_3
-\Lambda\cos{\theta}\e_1\wedge\e_2\nonumber\\
d(\e_1+i\e_2)&=&(-2e^{-i\theta}\omega
-ie^{-i\theta}\Lambda\e_3)\wedge(\e_1+i\e_2).
\end{eqnarray}
\end{prop}
{\bf Proof.} To establish this result we shall use the same strategy as in the proof of Proposition \ref{propGT1}. The calculations
are further complicated by the presence of $\theta$
but the main steps are as before: use the gauge freedom to set
$\widetilde{U}$ to a constant,
explore the Killing spinor equations to solve for the Maxwell potential $A$ and use the
Frobenius theorem to construct a triad of one--forms out of the
Killing spinors  defining a conformal structure on a three--manifold.

Using (\ref{spinor_cond}) and  (\ref{spinor_cond1})
we find
\begin{eqnarray*}
\nabla_a U^{-1}&=&2U^{-1}\cos{\theta} A_a+\alpha_A\hat{\beta}_{A'}
-\hat{\alpha}_A\beta_{A'}+\frac{1}{\Lambda}\phi_{AB}(e^{-i\theta}
\alpha^B\hat{\beta}_{A'}-e^{i\theta}\hat{\alpha}^B\beta_{A'}),\\
\nabla_a \widetilde{U}^{-1}&=&2\widetilde{U}^{-1}\cos{\theta}A_a+\Lambda
(\alpha_A\hat{\beta}_{A'}
-\hat{\alpha}_A\beta_{A'}),
\end{eqnarray*}
where $U, \widetilde{U}$ are defined by (\ref{UUbar}).
Use the gauge freedom in scalings of the spinors
to set $\widetilde{U}$ to a constant. This  gives an expression for $A$. Set
\[
T^a=\frac{1}{\sqrt{2}}(e^{-i\theta} \alpha^{A}\hat{\beta}^{A'}-e^{i\theta}\hat{\alpha}^{A}{\beta}^{A'}),
\]
and define a real one--form  $W$ by
$\alpha_A\hat{\beta}_{A'}=\sqrt{2}^{-1}e^{i\theta}(T_a+iW_a)$, so that
\[
g(W, W)=g(T, T)=(U\widetilde{U})^{-1}.
\]
We also define two real one--forms $\e_1, \e_2$ by
 $Z=f\sqrt{2}^{-1} (\e_1+i\e_2)$, where
$Z^a=\alpha^{A}\beta^{A'}$ and $f$ is some function.
Now introduce a local coordinate $\tau$ such that
$T^a\nabla_a=\p/\p\tau$ and so
\be
\label{form_of_T}
T=(U\widetilde{U})^{-1}(d\tau+\alpha)
\ee
for some one-form $\alpha$ which in general depends on $\tau$ .
Calculating $T^a\nabla_a U^{-1}$ yields
\[
\frac{\p}{\p\tau} U^{-1}= \sqrt{2}\cos\theta\;U^{-1}
(\widetilde{U}^{-1}-\Lambda U^{-1}).
\]
There are two cases to consider: If $U=\lambda\widetilde{U}=\const$, then $\phi_{AB}=0$, so $dA=0$ and without loss of generality we can
set $A=0$ in some gauge. Using an argument analogous to the one leading to
(\ref{hyperbolic_g}) we find that $g$ is a hyperbolic metric
with constant scalar curvature (this has $\Lambda>0$ in our conventions).
Otherwise we have
\[
\tau=\frac{1}{\mu}\;
\Big(\ln\frac{{\widetilde{U}}}{(U-\Lambda\widetilde{U})}\Big)+\ln{(c)},
\quad\mbox{where}\quad \mu=\sqrt{2}\widetilde{U}^{-1}\cos{\theta}, \;\;c=\const.\]
The Killing spinor equations give
\[
dZ= 2e^{i\theta} A\wedge Z-2UZ\wedge \bar{Y}-2\Lambda\widetilde{U}
Z\wedge Y,
\]
where $Y_a=\alpha_A\hat{\beta}_{A'}$.
Set $Z=f(\tau)Z_0$, where
$\dot{f}/f=\sqrt{2}\exp{(-i\theta)}\widetilde{U}^{-1}$ so that
\[
dZ_0=(\sqrt{2}e^{-i\theta}\widetilde{U}^{-1}\alpha+
i\Big(\frac{\sqrt{2}\Lambda \widetilde{U}}{\cos{\theta}}-
\sqrt{2}e^{-i\theta}U\Big) W)\wedge Z_0.
\]
We also find
\[
dT+idW= \sqrt{2}e^{-i\theta}(iU\; T\wedge W+ (U-\Lambda \widetilde{U})
|f|^2 Z_0\wedge\bar{Z}_0)
\]
and
\[
dW= \sqrt{2}\cos{\theta}(U\; T\wedge W-
(U-\Lambda \widetilde{U}) |f|^2 \e_1\wedge \e_2).
\]
Defining a one--form $\e_3$ by $W=g(\tau) \e_3$, where $g=g(0)\exp{(\mu\tau)}$ and substituting this
in the expression for $dW$ yields
\be
\label{de3}
d\e_3=\sqrt{2} \widetilde{U}^{-1}\cos{\theta}\; \alpha\wedge \e_3
-\sqrt{2}\beta \cos{\theta} \e_1\wedge \e_2,
\ee
where $\beta=\widetilde{U}|f(0)|^2/(c g(0))$ is a constant.
Similarly the expression for $dZ_0$ yields
\be
\label{de12}
d(\e_1+i\e_2) =(\sqrt{2} e^{-i\theta}\widetilde{U}^{-1}\alpha
+i\Big(\frac{2\Lambda\widetilde{U}}{\cos{\theta}}-2e^{-i\theta}U\Big)g\e_3)
\wedge(\e_1+i\e_2).
\ee
We now have to establish the dependence of $\alpha$ on $\tau$. The Killing spinor equations yield
\[
\nabla_{(a}T_{b)}= 2\cos{\theta} A_{(a} T_{b)}+\sqrt{2}^{-1}
(\widetilde{U}^{-1}+\Lambda U^{-1})\cos{\theta} g_{ab}.
\]
Therefore ${\cal L}_T h=\theta  \; h$, where $h_{ab}$ is the part of $g_{ab}$
orthogonal to $T^a$ and the last equality is valid modulo $T$.
Thus  $T_a$ is a conformal retraction. Moreover this retraction is ASD in the sense of \cite{Cald} as $dA$ is ASD. We further find
\[
{\cal L}_T T_a= \sqrt{2} \cos{\theta}(\widetilde{U}^{-1}-\Lambda U^{-1})T_a
-\sqrt{2} U^{-1}\sin{\theta}\; W_a.
\]
Finally using (\ref{form_of_T}) gives
\[
\alpha=-\omega+\Lambda \widetilde{U}^{2} g(0) \tan{\theta}\; e^{\mu\tau}\;
\e_3,
\]
where $\omega$ is some $\tau$--independent one--form orthogonal to $\p/\p\tau$.
To obtain equations (\ref{ewequations}) in the Proposition we  substitute this expression into (\ref{de3}) and (\ref{de12}), and make the following choices
for the so far unspecified constants
\[
\widetilde{U}=\frac{1}{\sqrt{2}}, \quad g(0)=c\Lambda, \quad
\beta=\sqrt{2}^{-1}\Lambda
\]
which is consistent if we also chose $(f(0))^2=(g(0))^2$.
Note that $c$ can also be chosen
arbitrarily by adding a constant to $\tau$. To obtain the formulae in the Proposition we set $c=\Lambda^{-1}$.
The metric $g$ is given by
\[
g=U\widetilde{U}( T^2+Z^2+|f|^2((\e_1)^2+(\e_2)^2)),
\]
where
\[
U=\frac{\widetilde{U}+c\widetilde{U}\Lambda e^{\mu\tau}}{c e^{\mu\tau}}.
\]
This, with our choice of constants, gives (\ref{last_metric}).
\koniec
{\bf Remark}. A three--dimensional Einstein--Weyl structure consists
of a conformal structure $[h]$ and a torsion--free connection $D$
such that
\[
D_i h_{jk}=\nu_i h_{jk}, \quad
R_{ij}+\frac{1}{2}\nabla_{(i}\nu_{j)}+\frac{1}{4}\nu_i\nu_j={\cal W}
h_{jk},\qquad
i, j, k=1, 2, 3,
\]
where $\nu$ is a one--form,
$\nabla_i$ and $R_{ij}$ are respectively
the Levi--Civita connection and the Ricci tensor of $h\in [h]$
and ${\cal W}$ is a function which can be read--off by taking a trace of both sides of the second equation. In \cite{DT99} it was shown that
\[
h=(\e_1)^2+(\e_2)^2+ (\e_3)^2, \quad \nu=-4\omega\cos{\theta}
-4\Lambda\sin{\theta} \e_3, \quad \Lambda=\const
\]
satisfies the Einstein--Weyl equations if the triad $(\e_1, \e_2, \e_3)$
satisfies (\ref{ewequations}). Moreover the Einstein--Weyl structure arising this way
is the most general symmetry reduction of hyper--K\"aher metric
in four dimension by a conformal symmetry.

\section{Euclidean Kastor--Traschen solutions}
\label{sectionthree}
In this section we shall drop the ASD condition on the Maxwell field so that
\[
F_{ab}=\phi_{AB}\vv_{A'B'}+\tilde{\phi}_{A'B'}\vv_{AB}.
\]
Thus the Killing spinor equations (\ref{spinor_cond0}) are
replaced by
\begin{eqnarray*}
\nabla_{AA'}\alpha_B+c_0 A_a\alpha_B+(c_1\phi_{AB}+c_2\vv_{AB})\beta_{A'}&=&0,\nonumber\\
\nabla_{AA'}\beta_{B'}+c_3 A_a\beta_{B'}+(c_5\tilde{\phi}_{A'B'}+ c_4\vv_{A'B'})\alpha_A&=&0,
\end{eqnarray*}
where an additional term involving $\tilde{\phi}_{A'B'}$ is present. We now impose the integrability conditions
(\ref{cc1})--(\ref{cc4}) and proceed as before, but use the Einstein--Maxwell condition
\[\Phi_{ABA'B'}=2\phi_{AB}\tilde{\phi}_{A'B'}.\] We find that
\[
c_0=c_3=\frac{1}{L}, \quad c_2=\frac{1}{2L}c_1, \quad c_5=-\frac{2}{c_1}, \quad c_4=-\frac{1}{Lc_1},
\]
where
\[
\Lambda=\frac{1}{2L^2}
\]
is a cosmological constant (which at this stage can be positive or negative if $L$ is real or imaginary respectively).
A constant rescaling of $\beta_{A'}$ can be used to set $c_1$ to any given non--zero constant.
To achieve a symmetric form of the equations
we replace $\beta_{A'}$ by $\sqrt{2}\beta_{A'}/c_1$  which results in $c_1=\sqrt{2}$. The final form of the
Killing spinor condition is
\begin{eqnarray}
\label{spinor_cond2}
\nabla_{AA'}\alpha_B+\frac{1}{L} A_a\alpha_B+\sqrt{2}(\phi_{AB}+\frac{1}{2L}\vv_{AB})\beta_{A'}&=&0,\nonumber\\
\nabla_{AA'}\beta_{B'}+\frac{1}{L} A_a\beta_{B'} -\sqrt{2}(\tilde{\phi}_{A'B'}+\frac{1}{2L}\vv_{A'B'})\alpha_A&=&0.
\end{eqnarray}

We conclude that the non-ASD case is more `rigid' than the ASD one. The Killing spinor equations
(\ref{spinor_cond}) with ASD Maxwell field contain one essential parameter $\theta$. If the Maxwell field is not ASD
(or SD) the integrability conditions fix all the parameters in terms of the cosmological constant.
Further analysis depends on the sign of the cosmological constant. If $\Lambda<0$, the resulting metric admits a Killing vector
$K_a=i(\hat{\alpha}_A\beta_{A'}+{\alpha}_A\hat{\beta}_{A'})$ and is given by a Riemannian analogue of the Caldarelli--Klemm solution \cite{CK,CK1}.
If $\Lambda>0$ the symmetry is not present in general, and the  metric is a  Riemannian version of the solutions obtained in \cite{GS, other_paper}.

In the proposition below we shall characterise Riemannian Kastor--Traschen solutions \cite{KS} as those where the ratio of norms of the spinors
$\alpha_A$ and $\beta_{A'}$ is a constant. We define two real non--zero functions $U, \widetilde{U}$ by
\[
U=(\varepsilon_{AB}\hat{\alpha}^A\alpha^B)^{-1}, \qquad \widetilde{U}=(\varepsilon_{A'B'}\hat{\beta}^{A'}\beta^{B'})^{-1}.
\]
as before. The gauge transformations $\alpha\rightarrow e^f\alpha, \beta\rightarrow e^f\beta$  where $f:M \longrightarrow \R$
result
in
\[
U\rightarrow e^{-2f} U, \quad \widetilde{U}\rightarrow e^{-2f}\widetilde{U}
\]
so that the ratio $U/\widetilde{U}$ is gauge invariant. In the rest of this section we shall assume $L=l\in\R$ and the cosmological constant is positive.
\begin{prop}
\label{propKT}
Let the Riemannian four--manifold $(M, g)$ admit a solution to the
Killing spinor equations {\em(\ref{spinor_cond2})} with $\Lambda>0$ such that the gauge invariant condition
\[
\frac{U}{\widetilde{U}}=\const
\]
holds. Then $(M, g)$ is Einstein--Maxwell with\footnote{The sign of the energy momentum tensor in this example is opposite to the one in
(\ref{lorentzian_EM}). This sign can be changed if desired as explained in the Introduction
by using the Maxwell field $F=2*d(({\cal U}+l^{-1}T)^{-1}dT)$.}
$F=2dA$ and
local coordinates $(x, y, z, T)$ can be chosen so
that
\be
g=\Big({\cal U}+\frac{1}{l}T\Big)^2(dx^2+dy^2+dz^2)+\Big(
{\cal U}+\frac{1}{l}T\Big)^{-2}dT^2, \quad
F=2\;d\Big(\frac{dT}{{\cal U}+l^{-1}T}\Big)
\label{KTmetric}
\ee
where ${\cal U}={\cal U}(x, y, z)$ satisfies the Laplace equation on $\R^3$
\[
\frac{\p^2 {\cal U}}{\p x^2}+\frac{\p^2 {\cal U}}{\p y^2}+\frac{\p^2 {\cal U}}{\p z^2}=0.
\]
\end{prop}
{\bf Proof.}
The Einstein--Maxwell equations with $\Lambda>0$ follow from the integrability conditions for (\ref{spinor_cond2}).
To find the local form of the metric first choose a gauge
\[
U\widetilde{U}=1.
\]
Set $L=l\in\R$. The Killing spinor equations and their conjugates can be used to find
\begin{eqnarray*}
\nabla_a(U^{-1})&=&-\frac{2}{l}A_a U^{-1}-\sqrt{2}\phi_A^BX_{BA'}-\frac{1}{l\sqrt{2}}X_a=0,\\
\nabla_a(\widetilde{U}^{-1})&=&-\frac{2}{l}A_a \widetilde{U}^{-1}-\sqrt{2}{\tilde{\phi}_{A'}}^{B'}X_{AB'}-\frac{1}{l\sqrt{2}}X_a=0.
\end{eqnarray*}
These equations imply
\[
U=\widetilde{U}=1, \quad A_aX^a=-\frac{1}{\sqrt{2}}, \quad
{{\tilde{\phi}_{A'}}}^{B'}X_{AB'}=\phi_A^BX_{BA'}.
\]
The expression for $A$ is found to be
\be
\label{form_of_A}
A_a=-\frac{l}{2\sqrt{2}}(E_a+\frac{1}{l}X_a),
\ee
where $E_a=2{\phi_A}^BX_{BA'}=X^bF_{ab}$. We also find
\be
\label{EX}
F= E\wedge X.
\ee
Further application of the Killing spinor equations (\ref{spinor_cond2}) gives
\begin{eqnarray}
\label{dzkx}
dZ&=&(-l^{-1}\sqrt{2}X-2l^{-1} A)\wedge Z,\nonumber\\
dK&=&(-l^{-1}\sqrt{2}X-2l^{-1} A)\wedge K,\\
dX&=&-2l^{-1}A\wedge X-\sqrt{2} F,\nonumber
\end{eqnarray}
and consequently
\[
{\cal L}_X Z=-\frac{\sqrt{2}}{l}Z, \quad {\cal L}_X K=-\frac{\sqrt{2}}{l}K, \quad X\wedge dX=0.
\]
The integrability conditions for the first two equations in (\ref{dzkx}) come down to
$d(X+\sqrt{2}A)=0$, so that locally
\be
\label{dxa}
X+\sqrt{2}A=d\gamma
\ee
for some function $\gamma$. Let $\tau$ be a local coordinate such that
$X^a\nabla_a=\p/\p\tau$. We find that $X(\gamma)=1$, so $\gamma=\tau+\tilde{\gamma}$,
where $\tilde{\gamma}$ is a function which does not depend on $\tau$. The
one--form dual to $X=\frac{\p}{\p\tau}$ is $X=2(d\tau+\Omega)$ for some one--form
$\Omega$. We can fix the falue of the constant $\tilde{\gamma}$ reabsorbing $d\tilde{\gamma}$ into the definition
of $\Omega$. Equations (\ref{dzkx}) now imply the existence of real local coordinates
$(x, y, z)$ such that
\[
K=\frac{1}{\sqrt{2}}e^{-\sqrt{2}\tau/l} dz, \quad Z=\frac{1}{2\sqrt{2}}e^{-\sqrt{2}\tau/l}(dx+idy)
\]
and combining (\ref{form_of_A}) with $X+\sqrt{2}A=d\tau$ yields $\Omega=lE/2$
so that the metric is
\[
g=e^{-2\sqrt{2}\tau/l}(dx^2+dy^2+dz^2)+2\Big(d\tau+l\frac{E}{2}\Big)^2.
\]
Using (\ref{dxa}) and (\ref{EX}) we calculate $dX=l dE=-(\sqrt{2})^{-1}F$ and
\begin{eqnarray*}
{\cal L}_XE&=&-\frac{1}{l\sqrt{2}}X\hook F=-\frac{1}{l\sqrt{2}}X\hook (E\wedge X)\\
&=&\frac{\sqrt{2}}{l} E
\end{eqnarray*}
as $X\cdot X=2$.
Therefore $E=e^{\sqrt{2}\tau/l}\omega$, where $\omega$ is a one--form independent on $\tau$.
The condition $X\wedge dX=0$ implies that $d\omega=0$ so that locally $\omega=d\phi$, where
$\phi=\phi(x, y, z)$ is some function. Using (\ref{dxa}) we find
\[
F=2e^{\sqrt{2}\tau/l}d\phi\wedge d\tau.
\]
Moreover
\[
*F=\frac{4\sqrt{2}}{l}*_3d\phi,
\]
where $*_3$ is the Hodge operator of the flat 3-metric. Therefore the Maxwell equation
$d*F=0$ implies that $\phi$ is harmonic on $\R^3$ and
\[
g=e^{-2\sqrt{2}\tau/l}(dx^2+dy^2+dz^2)+2\Big(d\tau+\frac{l}{2}e^{\sqrt{2}\tau/l}d\phi\Big)^2.
\]

To put the metric and the Maxwell field
in the form (\ref{KTmetric}) set
\[
T=l\frac{\phi}{\sqrt{2}}-l{e^{-\sqrt{2}\tau/l}}, \quad
{\cal U}=-\frac{\phi}{\sqrt{2}}.
\]
\koniec
The solution (\ref{KTmetric}) can be
obtained as an analytic continuation of the Kastor--Traschen
cosmological black holes \cite{KS}. This continuation
requires the sign of cosmological constant to change. \\ \\
{\bf Example.} Setting ${\cal U}=0$ in (\ref{KTmetric}) gives the hyperbolic space. Consider
${\cal U}=m/R$, where $m$ is a constant, and $R$ is the radial coordinate on $\R^3$ so that
the metric becomes
\be
\label{rnads}
g=\Big(\frac{m}{R}+\frac{T}{l}\Big)^2\Big(dR^2+R^2(d\theta^2+\sin^2{\theta} d\phi^2)\Big)
+\Big(\frac{m}{R}+\frac{T}{l}\Big)^{-2}dT^2.
\ee
This metric admits an isometry $(R, T)\rightarrow (c^{-1} R, cT)$ generated by the Killing vector
\[
{\cal K}=T\frac{\p}{\p T}-R\frac{\p}{\p R}.
\]
Introduce the coordinates $(s, r)$ adapted to this isometry by
\[
R=e^{-s/l}, \qquad T=l(r-m)e^{s/l}
\]
so that ${\cal K}(s)=0, {\cal K}(r)=1$. Let $\psi=\psi(s, r)$ be a
function such that
\[
d\psi=ds+\frac{l(r-m)}{Vr^2}dr,
\]
where
\[
V=\frac{r^2}{l^2}+\Big(1-\frac{m}{r}\Big)^2.
\]
The metric then takes the form
\be
\label{1instanton}
g=Vd\psi^2+V^{-1}dr^2+r^2(d\theta^2+\sin^2{\theta} d\phi^2).
\ee
It closely resembles the analytic continuation of the Reissner--Nordstr\"om--de Sitter
metric (\ref{RSDS}) described in the introduction in the extremal case
$|Q|=m$. The difference between these two solutions lies in the sign of the
cosmological constant. The extremal RNdS instanton with $\Lambda<0$
has been named the lukewarm instanton in \cite{Romans}. The conical singularities in the
metric are not present as the black--hole and the cosmological horizons have
the same Hawking temperatures, i.e. $V(r_1)=V(r_2)=0$ at these horizons
and $|V'(r_1)|=|V'(r_2)|$. This instanton has been interpreted
\cite{Romans, Ross-Man} as describing a pair creation of non-extreme black holes
in thermal equilibrium.

In our case $\Lambda>0$. At $r\rightarrow \infty$ the metric approaches the constant curvature hyperbolic space. The limit $r\rightarrow 0$
is singular.
This reflects the fact that
the metric (\ref{rnads}) is an analytic continuation of the
Lorentzian Reissner--Nordstr\"om--de Sitter space--time, where
the singularity is not hidden inside a horizon.
\\ \\
{\bf Example.} If $\Lambda=0$ and
\[{\cal U}=c+\sum_{m=1}^N\frac{a_m}{|{\bf x}-{\bf x}_m|},
\qquad a_1, \dots, a_N, c=\mbox{const}
\]
then (\ref{KTmetric}) becomes the Majumdar--Papapetrou Einstein--Maxwell multi instanton \cite{DH07}. The metric is asymptotically locally Robinson-Bertotti if $c=0$, or asymptotically flat if $c\neq 0$ and $T$ is periodic.
In \cite{DH07} it was shown how these instantons can be lifted to regular solitonic
solutions to ${\cal N}=2$ minimal five--dimensional supergravity. It remains to be seen
whether the solutions (\ref{KTmetric}) with non--zero $\Lambda$ can also
be uplifted to higher dimensions.
\section{Conclusions}
We have classified super--symmetric solutions of the minimal $N=2$ gauged
Euclidean supergravity in four dimensions, under the additional assumptions that the Maxwell field is anti--self--dual. The resulting metrics
are Einstein, have anti--self--dual Weyl curvature and are given in terms of solutions to three--dimensional
Einstein--Weyl equations. We have also found one class of examples
corresponding to non ASD Maxwell field. These examples are  Euclidean
analogs of Kastor--Traschen cosmological metrics. The solutions
constructed in the paper
provide new examples of Einstein metrics in four dimensions.
It remains to be seen whether they can be used to describe
cosmological black hole creations and in the context of Euclidean Quantum
Gravity.


TITLE: Show that the sequence $(s_n) = ((-1)^n n)$ is unbounded.
QUESTION [0 upvotes]: I need to prove that $(s_n)$ diverges. Since every convergent sequence is bounded, all I have to do is prove that $(s_n)$ is unbounded. But how?

REPLY [0 votes]: $|s_{n}| = n$. Imagine $s_{n}$ was bounded above and below by $M_{1}, M_{2}$ respectively, then $|s_{n}|< max(|M_{1}|, |M_{2}|)$, however choosing $n= \ceiling{max(|M_{1}|, |M_{2}|)}$ we get a contradiction. Thus $s_{n}$ is unbounded.


TITLE: Particles separation by shaking
QUESTION [1 upvotes]: In granulometry, say I have a tube with an heterogeneous mix of particles of different sizes (sand and stones, by instance). The tube is vertical in the usual gravity field and is placed in the air. If I randomly  shake the tube, the smallest particles will eventually be at the bottom, and the largest ones at the top. So the particles are now sorted by size. Why?
A friend of mine considers a volume of the bottom part of the tube has a greater density (or mass) than of the top's. And denser things go downward.
I consider however, by shaking the particles, small ones can fall through the space left between larger ones.
What is the correct explanation?

REPLY [1 votes]: There is a number of effects in play here.  I will assume a closed tube, because, if it is open or porous, the effects will change.  In general the following things occur:

If you shake it extremely vigorously, you get most of the material moving from end to end, and typically a complete mixing will occur.
If you shake it moderately vigorously, a convection-like current develops across the container, with particles rising in the centre and falling (relatively) at the walls.  This is called vibrofluidization, and is a mix of aerodynamic forces and granular collisions. (If you do this in a vacuum, the aerodynamic forces disappear, and the broad convection currents also disappear.)  Smaller and lighter (less dense) particles are more affected by the this current, so, while the net result is primarily mixing, denser particles and larger particles tend to drop towards the bottom and resist being entrained in the flow.
If you shake it less vigorously, (i.e. insufficiently for a convection-like current to be set up), vibro-compaction dominates.  (This is probably the region of the experiment you are wanting to look at.)  Here smaller particles do tend to fall into gaps when they emerge below the particles, or more likely, progressively wedge themselves lower.  Larger particles tend to allow gaps to form beside them, which are readily filled by smaller particles, and effectively float to the top when smaller particles fill gaps formed below them - buoyancy through a thousand wedges. At the surface when the large particle is slightly above the level of the sand, this effect is stronger, because there are fewer collisions from above pushing the large particle back down (which would normally limit the size of the gaps that form below the particle).  Overall, at this shaking level, particle-size effects dominate, and density plays a minor (though not negligible) role (so your answer is more correct than your friends).
If you shake even less vigorously, the shaking is insufficient to break (particularly Van Der Waals forces) that bind particles together, and the entire powder body moves as a single solid body.  Interestingly, the amplitude of vibration required to break these bonds increases as the particle size goes down.  Most sands are typically around 80 to 150 microns in size and flow quite well, but if they get below 40, they become very difficult to break apart.

Check out such other terms as vibro-fluidization, vibro-compaction, rheology, entrainment, acoustically enhanced fluidization, and Geldart types of particles.


TITLE: Let $G$ be a group of order $24$ having no normal subgroups of order $3$. Show that $G$ has four subgroups of order $6$.
QUESTION [4 upvotes]: Question

Let $G$ be a group of order $24$ having no normal subgroups of order $3$. Show that $G$ has four subgroups of order $6$.

Attempt
It is $24=3\cdot 2^3$ and $n_3\mid 2^3,\ n_3\equiv 1\pmod{3}\Rightarrow n_3=1,4$, $n_3\not=1$ from the hypothesis. Hence $n_3=4$. It is $n_2\mid 3$ and $n_2\equiv 1\pmod{2}\Rightarrow n_2=1,3$. If $n_2=1$ then there is a unique normal Sylow $2$-subgroup $P_2$ [and by taking the product with the $4$ Sylow $3$-subgroups we have the four subgroups of order $6$ needed.] {that's wrong}
Problem : If $n_2=3$ how could I proceed? EDIT:  if $n_2=1 ?$

REPLY [3 votes]: We know that $n_3=4$. Consider the conjugation action of $G$ on the four Sylow 3-subgroups of $G$, and let $I$ be the image of this action. So $I$ is a subgroup of $S_4$.
Since the Sylow subgroups are all conjugate in $G$, the action is transitive. So $|I|$ is divisible by 4. Also, since no Sylow subgroup can normalize another, the action of a Sylow 3-subgroup $P$ is a fixed point (i.e. $P$ itself), together with a 3-cycle. So $|I|$ is divisible by 3.
So $I$ is a subgroup of $S_4$ of order divisible by 12, and it must be $A_4$ or $S_4$.
Case 1. $I=A_4$. So the kernel $K$ of the action has order 2. Since $A_4$ has no subgroups of order $6$, the subgroups of order 6 in $G$ must contain $K$, and so they are the inverse images of the four subgroups of order $3$ in $I$. So there are four such subgroups in total, which are cyclic. (This is the case $n_2=1$.)
Case 2. $I=S_4$, so $I \cong G$, and as HallaSurvivor pointed out, there are also exactly four subgroups of order 6 (isomorphic to $S_3$)in this case. (This is the case $n_2=3$.)


text:
\begin{document}

\title{Exact Rényi entropies of $D$-dimensional harmonic systems}

\author{D. Puertas-Centeno, I. V. Toranzo and J. S. Dehesa}
\email[]{dehesa@ugr.es}
\affiliation{Departamento de F\'{\i}sica At\'{o}mica, Molecular y Nuclear, Universidad de Granada, Granada 18071, Spain}
\affiliation{Instituto Carlos I de F\'{\i}sica Te\'orica y Computacional, Universidad de Granada, Granada 18071, Spain}

\begin{abstract}
The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue \textit{per se} and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowest-lying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and Rényi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a $D$-dimensional quadratic potential, $D > 1$) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits.
In this work we determine the exact values of the R\'enyi uncertainty measures of the $D$-dimensional harmonic system for all ground and excited quantum states directly in terms of $D$, the potential strength and the hyperquantum numbers.

\end{abstract}

\maketitle

\section{Introduction}

The R\'enyi entropy of the probability density $\rho(\vec{r}), \vec{r}  =  (x_1 ,  \ldots  , x_D),$ which characterizes the quantum state of a $D$-dimensional physical system is defined \cite{renyi1,renyi2} as
\begin{equation}
\label{eq:renentrop}
R_{q}[\rho] =  \frac{1}{1-q}\log W_{q}[\rho], \quad 0<q<\infty, \,\, q \neq 1,
\end{equation}
where the symbol $W_{q}[\rho]$ denotes the frequency or entropic moment of order $q$ of the density given by
\begin{equation}
\label{eq:entropmom2}
W_{q}[\rho] = \int_{\mathbb{R}^D} [\rho(\vec{r})]^{q}\, d\vec{r}.
\end{equation}

These quantities completely characterize the density $\rho(\vec{r})$ \cite{romera_01,jizba2016} under certain conditions. They quantify numerous facets of the spreading of the quantum probability density $\rho(\vec{r})$, which include the intrinsic randomness (uncertainty) and the geometrical profile of the quantum system. The R\'enyi entropies are closely related to the Tsallis entropies \cite{tsallis} $T_{p}[\rho] = \frac{1}{p-1}(1-W_{p}[\rho]),  0<p<\infty,\, p\neq1$ by $T_{p}[\rho] = \frac{1}{1-p}[e^{(1-p)R_{p}[\rho]}-1]$. Moreover for the special cases $q=0,1,2,$ and $\infty $, the Rényi entropic power, $N_{q}[\rho]=e^{R_{q}[\rho]}$, is equal to $\text{the length of the support}, e^{-\langle \ln \rho \rangle}, \langle \rho \rangle^{-1}, \rho_{max}^{-1}$, respectively.
 Therefore, these $q$-entropies include the Shannon entropy \cite{shannon}, $S[\rho] = \lim_{p\rightarrow 1} R_{p}[\rho] = \lim_{p\rightarrow 1} T_{p}[\rho]$, and the disequilibrium, $\langle\rho\rangle =\exp(\textcolor{red}{-} R_{2}[\rho])$, as two important particular cases; in addition, they

The use of R\'enyi, Shannon and Tsallis entropies as measures of uncertainty allow a wider quantitative range of applicability than the Heisenberg-like measures which are based on the moments around the origin (so, including the standard or root-square-mean deviation). This permits, for example, a quantitative discussion of quantum uncertainty relations further beyond the conventional Heisenberg-like uncertainty relations \cite{hall,dehesa_sen12,bialynicki2,vignat,zozor2008,guerrero11,puertas2017}. The properties of the Rényi entropies and their applications have been widely analyzed; see e.g. \cite{aczel,leonenko,bialynicki3} and the reviews \cite{dehesa_sen12,jizba,jizba_2004b}.

In general, the R\'enyi entropies of quantum systems cannot be determined in an exact way, basically because the associated wave equation is generally not solvable in an analytical way. Even when the time-independent Schr\"{o}dinger equation is solvable, what happens for a small set of elementary potentials (zero-range, harmonic, Coulomb) \cite{albeverio,dong2011}, the exact determination of the R\'enyi entropies is a formidable task mainly because they are integral functionals of some special functions of applied mathematics \cite{nikiforov} (e.g., orthogonal polynomials, hypergeometric functions, Bessel functions,...) which control the wavefunctions of the stationary states of the quantum system. These integral functionals have not yet been solved for harmonic (i.e., oscillator-like) systems except for a few lowest-lying states (where the calculation is trivial) and, most recently, for the extreme Rydberg (i.e., highest-lying) \cite{aptekarev2016,dehesa2017,tor2016b} and pseudoclassical (i.e., the highest dimensional) \cite{tor2017b,puertas2017,temme2017} states of harmonic and Coulomb systems by means of sophisticated asymptotical techniques of orthogonal polynomials. This lack is amazing because harmonicity is the most frequent and useful approximation to study the quantum many-body systems, and the other two basic classes of uncertainty measures, the Heisenberg-like measures \cite{ray,drake,hey,assche,andrae,tarasov,zozor,suslov} and the Fisher information \cite{romera2005}, have been already calculated for all stationary states of the multidimensional harmonic system.\\

In this work we determine the exact values of the R\'enyi uncertainty measures of the $D$-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) for all ground and excited quantum states directly in terms of $D$, the potential strength and the hyperquantum numbers which characterize the states.

This is a far more difficult problem than the Heisenberg-like and Fisher information cases, both analytically and numerically. The latter is basically because a naive numerical evaluation using quadratures is not convenient due to the increasing number of integrable singularities when the principal hyperquantum number is increasing, which spoils any attempt to achieve reasonable accuracy even for rather small hyperquantum numbers \cite{buyarov}.

The structure of the manuscript is the following. In section \ref{sec:basics} the wavefunctions and the probability densities of the stationary states of the $D$-dimensional harmonic (oscillator-like) system are briefly described in both position and momentum spaces. In section \ref{sec:renyi} the R\'enyi entropies for all the ground and excited states of this system are determined in an analytical way by use of a recently developed methodology \cite{amc2013}. Finally some conclusions and open problems are given.

\section{The $D$-dimensional harmonic problem}
\label{sec:basics}
In this section we summarize  the quantum-mechanical $D$-dimensional problem corresponding to the harmonic oscillator potential
\begin{equation}
V(r) = \frac{1}{2}k(x_{1}^{2}+\ldots + x_{D}^{2}) = \frac{1}{2}kr^{2},
\end{equation}
and we give the probability densities of the stationary quantum states of the system in both position and momentum spaces. The stationary bound states of the system, which are the physical solutions of the Schr\"{o}dinger equation
\begin{equation}\label{schrodinger}
\left( -\frac{1}{2} \vec{\nabla}^{2}_{D} + V(r)\right) \Psi \left( \vec{r} \right) = E \Psi \left(\vec{r} \right),
\end{equation}
(we use atomic units throughout the paper) where $\vec{\nabla}_{D}$ denotes the $D$-dimensional gradient operator, are well known \cite{gallup,louck60a,yanez1994} to be characterized by the energies
\begin{equation}
\label{HOEL}
E_{N} = \left(N + \frac{D}{2}\right) \omega
\end{equation}
where
\[
\omega = \sqrt{k}, \quad N = \sum_{i=1}^{D}n_{i} \quad \text{with} \quad n_{\textcolor{red}{i}}=0,1,2,\ldots
\]
The corresponding eigenfunctions can be expressed as
\begin{equation}
\label{HOEF}
\psi_{N}(\vec{r}) = \mathcal{N} e^{-\frac{1}{2}\alpha(x_{1}^{2}+\ldots+x_{D}^{2})}H_{n_{1}}(\sqrt{\alpha}\, x_{1})\cdots H_{n_{D}}(\sqrt{\alpha}\, x_{D}), \quad \alpha = k^{\frac{1}{4}}
\end{equation}
where $\vec r\in\mathbb R^D$ and $\mathcal{N}$ stands for the normalization constant
\[
\mathcal{N} = \frac{1}{\sqrt{2^{N}n_{1}!n_{2}!\cdots n_{D}! }}\left(\frac{\alpha}{\pi}\right)^{D/4},
\]
and $H_{n}(x)$ denotes the Hermite polynomials of degree $n$ orthogonal with respect the weight function $\omega(x) = e^{-x^{2}}$ in $(-\infty, \infty)$.\\
Then, the associated quantum probability density in position space is given by
\begin{equation}
\label{HOPD}
\rho_{N}(\vec{r}) = |\psi_{N}(\vec{r})|^{2} = \mathcal{N}^{2} e^{-\alpha(x_{1}^{2}+\ldots+x_{D}^{2})}H_{n_{1}}^{2}(\sqrt{\alpha}\, x_{1})\cdots H_{n_{D}}^{2}(\sqrt{\alpha}\, x_{D}),
\end{equation}
and the density function in momentum space is obtained by squaring the Fourier transform of the position wavefunction, obtaining
\begin{align}
\label{HOMPD}
\gamma_{N}(\vec{p}) & = \mathcal{\tilde{N}}^{2} e^{-\frac{1}{\alpha}(p_{1}^{2}+\ldots+p_{D}^{2})}H_{n_{1}}^{2}\left(\frac{ p_{1}}{\sqrt{\alpha}}\right)\cdots H_{n_{D}}^{2}\left(\frac{ p_{D}}{\sqrt{\alpha}}\right)

= \alpha^{-D}\rho_{N}\left(\frac{\vec{p}}{\alpha}\right)
\end{align}
where $\vec p\in\mathbb R^D$ and the normalization constant is
\[
\mathcal{\tilde{N}}  = \frac{1}{\sqrt{2^{N}n_{1}!\cdots n_{D}! }}\left(\frac{1}{\pi\alpha}\right)^{D/4}.
\]

\section{R\'enyi entropies of the harmonic system}
\label{sec:renyi}
Let us now determine the R\'enyi entropy of the $D$-dimensional harmonic system according to Eqs. \eqref{eq:renentrop}-\eqref{eq:entropmom2} by
\begin{align}
\label{HORE}
R_{q}[\rho_{N}] &= \frac{1}{1-q}\log \int_{-\infty}^{\infty} dx_{1}\ldots \int_{-\infty}^{\infty} dx_{D} \, [\rho_{N}(\vec{r})]^{q} \nonumber \\
& =  \frac{1}{1-q}\log\left( \mathcal{N}^{2q}\int_{-\infty}^{\infty} e^{-\alpha q x_{1}^{2}}|H_{n_{1}}(\sqrt{\alpha}\, x_{1})|^{2q} \, dx_{1} \cdots \int_{-\infty}^{\infty} e^{-\alpha q x_{D}^{2}}|H_{n_{D}}(\sqrt{\alpha}\, x_{D})|^{2q}\, dx_{D} \right)
\end{align}
where we have used Eq. \eqref{HOPD}. To calculate these $D$ integral functionals of Hermite polynomials we will follow the 2013-dated technique (only valid for $q\in\mathbb{N}$ other than unity) \cite{srivastava,niukkanen,amc2013} to evaluate similar integral functionals of hypergeometric orthogonal polynomials by means of multivariate special functions. To do so, first we express the Hermite polynomials in terms of the Laguerre polynomials (see e.g., \cite{olver}) as
\begin{eqnarray}
\label{HfL}
H_{2n}(x) &=& (-1)^{n} 2^{2n}n!L_{n}^{-\frac{1}{2}}(x^{2}), \nonumber \\
H_{2n+1}(x) &=& (-1)^{n} 2^{2n+1}n!xL_{n}^{\frac{1}{2}}(x^{2}),
\end{eqnarray}
which allows to write

\begin{equation}
\label{HpfL}
H_{n}(\sqrt{\alpha}x) ^{2q} = A_{n,q}(\nu) \alpha^{q\nu}x^{2q\nu} L_{\frac{n-\nu}{2}}^{(\nu-\frac{1}{2})}(\alpha x^{2}) ^{2q},
\end{equation}
with the constant

\[
A_{n,q} (\nu) = 2^{2qn}\left[\Gamma\left(\frac{n-\nu}{2}+1\right) \right]^{2q}
\]
and the paramater $\nu=0(1)$ for even(odd) $n$; that is, $\nu=\frac12\left(1-(-1)^{n}\right).$

\\
Following the same steps as in \cite{amc2013}, after the change of variable $t_{i}=\alpha q x_{i}^{2}$ in \eqref{HORE}, one obtains the following linearization relation for the $(2q)$-th power of the Hermite polynomials

	\begin{equation}
		\label{linforH2}
		H_{n}\left(\sqrt \alpha x\right)^{2q} = A_{n,q}(\nu)q^{-q\nu}\sum_{j=0}^{\infty}\frac{1}{(-1)^{}2^{2j } j!}c_{j}\left(q\nu,2q,\frac{1}{q},\frac{n-\nu}{2},\nu-\frac{1}{2},-\frac{1}{2} \right) H_{2j}(\sqrt{\alpha q}x),
		\end{equation}

with

\begin{align}
\hspace{-3cm}c_{j}&\left( q\nu,2q,\frac{1}{q},\frac{n-\nu}{2},\nu-\frac{1}{2},-\frac{1}{2} \right)= \nonumber \\
& = \left(\frac{1}{2}\right)_{q\nu} \binom{\frac{n+\nu-1 }{2}}{\frac{n-\nu}{2}}^{2q} F_{A}^{(2q+1)}\left( \begin{array}{cc}
							q\nu+\frac{1}{2} ; \overbrace{\frac{\nu-n }{2}, \ldots, \frac{\nu - n}{2}}^{2q}, -j & \\[-3.5em]
																			&; \underbrace{\frac{1}{q}, \ldots, \frac{1}{q}}_{2q},1\\[-3.5em]
							\underbrace{\nu + \frac{1}{2}, \ldots, \nu+\frac{1}{2}}_{2q},\frac{1}{2} & \\
							\end{array}\right),\nonumber\\
\end{align}
where $(z)_a = \frac{\Gamma(z+a)}{\Gamma(z)}$ is the known Pochhammer's symbol and  $F_{A}^{(2q+1)}(\frac{1}{q}, \ldots, \frac{1}{q},1)$ is the Lauricella function of type A of $2q+1$ variables given by

\begin{align}
\label{HOLF}
F_{A}^{(2q+1)}\left( \begin{array}{cc}
							q\nu+\frac{1}{2} ; \frac{\nu-n }{2}, \ldots, \frac{\nu - n}{2},-j & \\[-3.5em]
																			&; \frac{1}{q}, \ldots, \frac{1}{q},1\\[-3.5em]
							\nu+ \frac{1}{2}, \ldots, \nu+\frac{1}{2},\frac{1}{2} & \\
							\end{array}\right) &=\nonumber\\
							&\hspace{-7cm}=  \sum_{k_{1}, \ldots, k_{2q}, k_{2q+1}=0 }^{\infty} \frac{\left(q\nu+\frac{1}{2}\right)_{k_{1}+\ldots k_{2q}+ k_{2q+1}} (\frac{\nu_-n }{2})_{k_{1}} \cdots (\frac{\nu-n }{2})_{k_{2q}}(-j)_{ k_{2q+1}} }{(\nu+ \frac{1}{2})_{k_{1}} \cdots (\nu + \frac{1}{2})_{k_{2q}}\left(\frac{1}{2}\right)_{ k_{2q+1}} } \frac{\left(\frac{1}{q}\right)^{k_{1}} \cdots \left(\frac{1}{q}\right)^{k_{2q}}}{k_{1}!\cdots k_{2q}!  k_{2q+1}!} ,
\end{align}
 Now, the combination of Eqs. \eqref{HORE} and \eqref{linforH2} together with the orthogonalization condition of the Hermite polynomials $H_{n}(x)$ (with which one realizes that all the summation
terms vanish except the one with $i=0$), allows one to write the exact Rényi entropy of the harmonic system as
\begin{align}
\label{HORE1}
R_{q}[\rho_{N}] &=\frac{1}{1-q}\log \left[\mathcal{N}^{2q} \left(\frac{\pi}{\alpha}\right)^{\frac{D}{2}}q^{-\frac{D}{2}}\prod_{i=1}^{D}q^{-q\nu_{i}}A_{n_{i},q}(\nu_{i})\, c_{0}\left( q\nu_{i},2q,\frac{1}{q},\frac{n_{i}-\nu_{i}}{2},\nu_{i}-\frac{1}{2},-\frac{1}{2} \right)\right]  \nonumber \\

&\hspace{-1cm}= \frac D2 \log\left[\frac{\pi}{\alpha}\right]+\frac{1}{q-1}\log \left[2^{qN} q^\frac D2  \right] +\frac{1}{1-q}\sum_{i=1}^{D}\log\left[\frac{A_{n_{i},q}(\nu_{i})}{q^{q\nu_{i}}\Gamma(n_i+1)^q}\, c_{0}\left( q\nu_{i},2q,\frac{1}{q},\frac{n_{i}-\nu_{i}}{2},\nu_{i}-\frac{1}{2},-\frac{1}{2} \right) \right]
\end{align}
with

\begin{align}
c_{0}&\left( q\nu,2q,\frac{1}{q},\frac{n-\nu}{2},\nu-\frac{1}{2},-\frac{1}{2} \right) = \left(\frac{1}{2}\right)_{q\nu} \binom{\frac{n+\nu-1 }{2}}{\frac{n-\nu}{2}}^{2q}\, \mathfrak F_q(n),\nonumber\\
\end{align}
where the symbol $\mathfrak F_q(n)$ denotes the following Lauricella function  of $2q$ variables

\begin{align}
\label{HOLF2}
\mathfrak F_q(n)&\equiv
F_{A}^{(2q+1)}\left( \begin{array}{cc}
							q\nu+\frac{1}{2} ; \frac{\nu-n }{2}, \ldots, \frac{\nu - n}{2},0 & \\[-3.5em]
																			&; \frac{1}{q}, \ldots, \frac{1}{q},1\\[-3.5em]
							\nu+ \frac{1}{2}, \ldots, \nu+\frac{1}{2},\frac{1}{2} & \\
							\end{array}\right) =
F_{A}^{(2q)}\left( \begin{array}{cc}
							q\nu+\frac{1}{2} ; \frac{\nu-n }{2}, \ldots, \frac{\nu - n}{2} & \\[-3.5em]
																			&; \frac{1}{q}, \ldots, \frac{1}{q}\\[-3.5em]
							\nu + \frac{1}{2}, \ldots, \nu +\frac{1}{2} & \\
							\end{array}\right) &\nonumber\\
							&=  \sum_{j_{1}, \ldots, j_{2q}=0 }^{\infty} \frac{\left(q\nu+\frac{1}{2}\right)_{j_{1}+\ldots j_{2q}} (\frac{\nu-n }{2})_{j_{1}} \cdots (\frac{\nu-n }{2})_{j_{2q}} }{(\nu + \frac{1}{2})_{j_{1}} \cdots (\nu+ \frac{1}{2})_{j_{2q}} } \frac{\left(\frac{1}{q}\right)^{j_{1}} \cdots \left(\frac{1}{q}\right)^{j_{2q}}}{j_{1}!\cdots j_{2q}! }&
							\nonumber\\
							&=  \sum_{j_{1}, \ldots, j_{2q}=0 }^{\frac{n-\nu}2} \frac{\left(q\nu+\frac{1}{2}\right)_{j_{1}+\ldots j_{2q}} (\frac{\nu-n }{2})_{j_{1}} \cdots (\frac{\nu-n }{2})_{j_{2q}} }{(\nu + \frac{1}{2})_{j_{1}} \cdots (\nu + \frac{1}{2})_{j_{2q}} } \frac{\left(\frac{1}{q}\right)^{j_{1}} \cdots \left(\frac{1}{q}\right)^{j_{2q}}}{j_{1}!\cdots j_{2q}! }.
\end{align}
Note that, as $\frac{\nu-n}{2}$ is always a negative integer number, the Lauricella function simplifies to a finite sum. In the following, for convenience, we use the notation $N_O=\sum_{i=1}^D\nu_i$, which is the amount of odd numbers $n_i$ and, thus, $N_E=D-N_O$ gives the number of the even ones. Then simple algebraic manipulations allow us to rewrite Eq. \eqref{HORE1} as

\begin{eqnarray}
\label{HORE3}\nonumber
R_{q}[\rho_{N}]
&=&    -\frac D2 \log\left[\alpha\right]+\mathcal K_q\,D+\overline{\mathcal K}_q\,N_O+\frac {q}{q-1}\sum_{i=1}^D(-1)^{n_i}\log\left[\left(\frac{n_i+1}{2}\right)_{\frac12}  \right]+\frac{1}{1-q}\sum_{i=1}^D\log\left[\mathfrak F_q(n_i)\right], \nonumber\\
\end{eqnarray}
where $\mathcal K_q=\frac{\log[\pi^{q-\frac12}\,q^\frac12]}{q-1}$ and $\overline{\mathcal K}_q=\frac{1 }{1-q}\log \left[\frac{4^{q}\,\Gamma\left(\frac12+q\right)}{\pi^{\frac{1}{2}}\,q^{q}}\right]$.
This expression allows for the analytical determination of the Rényi entropies (with positive integer values of $q$) for any arbitrary state of the multidimensional harmonic systems.

Finally, for the ground state (i.e., $n_i=0,\,i=1,\cdots, D$; so, $N=0$) the general Eq. \eqref{HORE3} boils down to ,
	 	 \begin{equation}
	 	 R_q[\rho_N]=\frac D2\log\left[\frac {\pi\, q^{\frac1{q-1}}}{\alpha}\right].
	 	 \end{equation}

In fact, this ground state R\'enyi entropy holds for any $q>0$ as one can directly derive from Eq. \eqref{HORE}. Taking into account that the momentum density is a re-scaled form of the position density, we have the following expression for the associated momentum R\'enyi entropy,
\begin{eqnarray}
\label{Remsp}
R_{\tilde q}[\gamma_N] &=&  \frac D2 \log\left[\alpha\right]+\mathcal K_{\tilde q}\,D+\overline{\mathcal K}_{\tilde q}\,N_O+\frac {\tilde q}{\tilde q-1}\sum_{i=1}^D(-1)^{n_i}\log\left[\left(\frac{n_i+1}{2}\right)_{\frac12}  \right]+\frac{1}{1-\tilde q}\sum_{i=1}^D\log\left[\mathfrak F_{\tilde q}(n_i)\right],\nonumber\\
\end{eqnarray}
($\tilde q\in \mathbb{N}$). Although Eqs. \eqref{HORE3} and \eqref{Remsp} rigorously hold for $q\not=1$ and $q\in\mathbb N$ only, it seems reasonable to conjecture its general validity for any $q>0, \,q\not=1$ provided the formal existence of a generalized function $\mathfrak F_q(n)$. If so, we obtain the general expression for the position-momentum uncertainty Rényi entropic sum as
\begin{eqnarray}\nonumber
R_{q}[\rho_N]+R_{\tilde q}[\gamma_N] &=&  (\mathcal K_{q}+\mathcal K_{\tilde q})\,D+(\overline{\mathcal K}_{q}+\overline{\mathcal K}_{\tilde q})\,N_O+\left(\frac {q}{ q-1}+\frac {\tilde q}{\tilde q-1}\right)\sum_{i=1}^D(-1)^{n_i}\log\left[\left(\frac{n_i+1}{2}\right)_{\frac12}  \right]
\\
&+&\frac{1}{1- q}\sum_{i=1}^D\log\left[\mathfrak F_{q}(n_i)\right]+\frac{1}{1-\tilde q}\sum_{i=1}^D\log\left[\mathfrak F_{\tilde q}(n_i)\right]
\end{eqnarray}
which verifies the R\'enyi-entropy-based uncertainty relation of Zozor-Portesi-Vignat \cite{zozor2008} when $\frac1q+\frac1{\tilde q}\ge2$ for arbitrary quantum systems.
In the conjugated case $\tilde q=q^*$ such that $\frac1q+\frac1{q^*}=2$, one obtains
\begin{eqnarray}\nonumber
R_{q}[\rho_N]+R_{q^*}[\gamma_N] &=&  D\log\left(\pi q^{\frac1{2q-2}}{q^*}^{\frac{1}{2q^*-2}}\right)+(\overline{\mathcal K}_{q}+\overline{\mathcal K}_{ q^*})\,N_O\\
&+&\frac{1}{1- q}\sum_{i=1}^D\log\left[\mathfrak F_{q}(n_i)\right]+\frac{1}{1- q^*}\sum_{i=1}^D\log\left[\mathfrak F_{ q^*}(n_i)\right].
\end{eqnarray}
Let us finally remark that the first term corresponds to the sharp bound for the general Rényi entropy uncertainty relation with conjugated parameters
\begin{equation}\nonumber
R_{q}[\rho_N]+R_{q^*}[\gamma_N] \ge D\log\left(\pi q^{\frac1{2q-2}}{q^*}^{\frac{1}{2q^*-2}}\right)
\end{equation}
of Bialynicki-Birula \cite{bialynicki2} and Zozor-Vignat \cite{vignat}.

\section{Conclusions}

In this work we have explicitly calculated the R\'enyi entropies, $R_q [\rho_N]$ ($q\in \mathbb{N}$), for all the
quantum-mechanically allowed harmonic states in terms of the Rényi index $q$, the spatial dimension $D$, the oscillator strength $\alpha$, as well as
the hyperquantum numbers, $\{n_{i}\}_{i=1}^{D}$, which characterize the corresponding state's wavefunction. To do that we have used the harmonic wavefunctions in Cartesian coordinates, which can be expressed in terms of a product of $D$ Hermite polynomials and exponentials. So, the R\'enyi entropies of the quantum states boil down to $D$ entropy-like functionals of Hermite polynomials. Then we have determined these integral functionals by taking into account the close connection between the Hermite and Laguerre polynomials and the Srivastava-Niukkanen linearization method for powers of Laguerre polynomials. The final analytical expression of the Rényi entropies with positive integer index $q$ in both position and momentum spaces is given in a compact way by use of a Lauricella function of type A. It remains as an open problem, the extension of this result to R\'enyi entropies for any real value of the parameter $q$. The latter requires a completely different approach, still unknown to the best of our knowledge.

\section*{Acknowledgments}
This work has been partially supported by the Project FQM-207 of the Junta de Andaluc\'ia and the MINECO-FEDER grants FIS2014-54497P and FIS2014-59311P. I. V. Toranzo acknowledges the support of ME under the program FPU.
\\
Author contribution statement:
all authors have contributed equally to the paper.