text:
\begin{document}

\begin{abstract}
  In this paper we present a spectral collocation method for the fast evaluation
of the Landau collision operator for plasma physics, which
allows us to obtain spectrally accurate numerical solutions. The method is
inspired by the seminal work \cite{PRT2}, but it is specifically designed for Coulombian interactions, taking into account
the particular structure of the operator. It allows us to reduce the
number of discrete convolutions to provide an approximation
of the Landau operator.  Then,  we show that the method preserves the total
mass whereas momentum and energy are approximated with spectral
accuracy. Numerical results for the Landau equation in three dimensions in velocity
space are presented to illustrate the efficiency of the present approach.
\end{abstract}

\date{\today}
\maketitle

\vspace{0.1cm}
\noindent
{\it Keywords}: Landau equation, spectral collocation methods, plasma
physics applications.
\\

\vspace{0.1cm}
\noindent
{\it 2010 MSC}: 35R09, 65M70, 65N35.

\tableofcontents

\section{Introduction}
\label{sec:introduction}
\setcounter{equation}{0}

The Landau or Fokker-Planck-Landau equation is a common kinetic model
applied to collisional  plasmas. More precisely, this mathematical model describes binary collisions between charged particles with
long-range Coulomb interaction and is represented by a nonlinear
partial integro-differential equation written as,
\be
\label{eq:0}
\frac{\partial f}{\partial t} + \bv.\nabla_\bx f = \cC(f,f),
\ee
where the collision operator $\cC(f,f)$ is given by
\be
\cC(f,f) = \nabla_\bv\cdot\int_{\R^3}\bA(\bv-\bv^*)\biggr[
  \nabla_\bv f(\bv)\,f(\bv^*) \,-\, \nabla_{\bv^*}f(\bv^*)\,f(\bv)\biggr] \dD
  \bv^*,
\label{eq:1}
  \ee
with the matrix $\bA(\bz)$,
\begin{equation}
\label{def:A}
\bA(\bz) = \,|\bz|^{\gamma +2}\,\left( \bI - \frac{\bz\otimes \bz}{|\bz|^2}\right)\,,
\end{equation}
where $\bI$ represents the identity matrix in dimension three. The function $f$ represents the density of a gas in
phase space at positions $\bx$ and velocities $\bv$, it is nonnegative
and integrable together with its moments in velocity
up to the second order $f\in L^1((1+|\bv|^2)\dD \bx\dD\bv)$. Actually
it is worth to mention that different values of $\gamma$ lead to usual
classification in hard potentials $\gamma > 0$, Maxwellian molecules
$\gamma = 0$, or soft potentials  $\gamma < 0$. However, this latter
case is of primary importance in plasma physics since it
involves the Coulombian case when $\gamma = -3$. Furthermore, the algebraic structure of the operator is similar to the Boltzmann
one for rarefied gas dynamics, this leads to physical properties such as the conservation of mass, impulsion, and energy
\begin{eqnarray*}
\int_{\R^3} \cC(f,f)(\bv)\left(
\begin{array}{c}
1 \\
\bv \\
|\bv|^2
\end{array}
\right)
\dD \bv \,=\, 0,
\end{eqnarray*}
and the decay of the kinetic entropy $\cH[f](t)$,
\be
\label{eq:H}
\frac{\dD\,\cH[f]}{dt} \,=\, \frac{\dD}{\dD t}\int_{\R^3} f(\bv)\,\ln
(f(\bv))\,\dD \bv\,\leq\, 0.
\ee
Finally, the equilibrium states of the Landau operator, {\it i.e.}
functions $f$ such that $\cC(f,f) = 0$, are given by Maxwellian distribution functions:
\begin{equation}
  \label{eq:M}
\cM_{\rho, \bu,T}(\bv) \,=\, \frac{\rho}{(2\,\pi\, v_{th}^2)^{3/2}} \,e^{-\frac{|\bv-\bu|^2}{2\,v_{th}^2}},
\end{equation}
where $\rho$ is the total mass, $\bu$ the mean velocity,
\be
\label{def:rho-u}
\rho \,=\, \int_{\R} f(\bv)\,\dD \bv, \quad \rho\,\bu \,=\, \int_{\R} f(\bv)\,\bv\,\dD \bv,
\ee
and $v_{th}$ represents the thermal velocity, which depends on the
temperature $T$,
\be
\label{def:T}
3\,\rho \, T \, =\, \int_{\R} f(\bv) |\bv-\bu|^2 \,\dD \bv,
\ee
as $v_{th} = \sqrt{{k_B\,T}/{m}}$, where  $m$ represents the mass of one
particle and $k_B$ is the Boltzmann constant. Notice that these
quantities are observables and can be deduced directly from the moments
of the distribution function with respect to the velocity.

To investigate the long
time behavior of the solution, the relative entropy $\cH[f|\cM_{\rho, \bu,T}]$ plays
a major role, it is defined as
\be
\label{h:rel}
\cH[f|\cM_{\rho, \bu,T}] \,:=\,  \cH[f|] - \cH[\cM_{\rho, \bu,T}] = \int_{\R^3} f
\,\log\left(\frac{f}{\cM_{\rho, \bu,T}}\right)\,\dD \bv.
\ee
It allows to measure the distance between the solution $f$ and the
equilibrium. Indeed, thanks to the Csisz\'ar-Kullback inequality, we have
$$
\| f - \cM_{\rho, \bu,T} \|_{L^1}^2 \,\leq \, C_{CK}\, \cH[f|\cM_{\rho, \bu,T}].
$$

On the one hand, the collision operator (\ref{eq:1}) is classically obtained
as a remedy of the
Boltzmann operator for a sequence of scattering cross sections which
converge in a convenient sense to a delta function at zero scattering
angle. In Coulomb collisions small
angle collisions play a more important role than collision
resulting in large velocity changes. The original derivation of the
equation is based on this idea is due
to Landau \cite{Landau}.  Depending on one's taste and notion of rigor several mathematical
derivation of the equations have been performed, we mention here
the works of Rosenbluth et al.~\cite{RMJ},   Bobylev \cite{bobylev1975}, Arsen'ev and Buryak~\cite{arsen:91}, Degond and
Lucquin-Desreux~\cite{degond} and
Desvillettes~\cite{desvillettes}. We refer to \cite{vill:new:00}  for  a  review of the main
mathematical aspects related to this issue.

On the other hand, Arsene'v and Peskov
\cite{arsenev0} have established the existence of weak solutions for
short time in the spacially homogeneous case for the Coulomb
potential. Later a global existence result of renormalized
solution with a defect measure has been obtained by Alexandre and
Villani \cite{alex0,villani0} in the space dependance case and for an
initial data with a finite energy.

Concerning the approximation of the Landau equation
\eqref{eq:0}-\eqref{eq:1}, various numerical methods have been studied. Depending on how to discretize the velocity space, these
methods can be categorized into two classes,
\begin{itemize}
  \item deterministic  methods based on
    finite difference schemes on a grid in velocity;
    \item  probabilistic
algorithms also known as particles methods.
\end{itemize}

In contrast with the Boltzmann equation where Monte Carlo methods
plays a major role in numerical simulations,  the extension to long range forces, in particular
Coulomb interactions, is more challenging and still requires a lot of attention \cite{BP, RRCD}. Most of the particle methods for Coulomb interaction have been derived more on a physical
intuition basis and not directly from the Landau equation \cite{Nanbu,BP}.  A detailed discussion about this is beyond the aims
of the present paper and we refer the reader to \cite{Nanbu,BP, PBM,
  RRCD}  for a more complete treatment and to \cite{PBM, DMP} for  reviews.

On the other hand, several deterministic numerical approaches have
been considered  for
the Landau collision operator  \cite{berezin,bobylev0,buet0,bcdl,
  degond0,DK,epp,lemou, lucquin,pota2,chacon1, chacon2, Zaitsev}. However, due to the computational complexity of the equation, which is essentially caused by
the large number of variables and the three-fold collision integral,
many papers have been devoted to simplified problems as for the isotropic case \cite{bobylev0} or
for cylindrically symmetric problems in \cite{pekker, KK}. Later from the late 1990s, a considerable amount of attention
has been directed towards the full Landau equation.  The construction
of conservative and entropic finite difference schemes for the space
homogeneous case has been proposed by Degond and Lucquin-Desreux in
\cite{degond0} and Buet and  Cordier \cite{buet0}.  These  schemes are
built in such a way that the main physical properties are conserved at a discrete level. Positivity of the
solution and discrete entropy inequality are also satisfied.
Unfortunately, the direct implementation of such
schemes for space non homogeneous computations is very expensive
since the computational cost increases roughly in proportion to
the square of the number of parameters used to represent the
distribution function in the velocity space.  Thus several fast approximated algorithms to reduce the
computational complexity of these methods, based on
multipole expansions \cite{lemou} or multigrid
techniques \cite{buet0} have been proposed.
Although these fast schemes are able to preserve the most
relevant  physical properties, the range of applications
seems limited and the degree of accuracy of such
approaches has not been studied.  Most of these results are provided when we neglect the
space variable in \eqref{eq:0},  we   refer to \cite{CF, DDFT}
for a direct implementation of such schemes taking into account and
space variable and self-consistent interactions in  one \cite{CF} or
two dimensions \cite{DDFT} in the
physical space and in three dimensions in velocity space.

In the meantime, a different approach based on spectral methods, has been
proposed for the Boltzmann \cite{PR, PP} and
Landau \cite{PRT1, PRT2, FP} collision operators.
In that case,  the spectral scheme permits
to obtain spectrally accurate solutions with a reduction of
the quadratic cost $N^2$ to $N\,\log_2\,N$, where $N$ is
the total number of unknowns in the velocity space. The lack of discrete
conservations in the spectral scheme (mass is preserved, whereas momentum
and energy are approximated with spectral accuracy) is compensated by
its higher accuracy and efficiency. In particular the scheme allows
easily the implementation of grid-refinement techniques in the velocity space.

A detailed comparison of the spectral method with the
schemes proposed in \cite{bcdl,lemou} has been done in
\cite{Filbet}. For the same degree of freedom $N$, the computational cost of spectral
algorithm is much higher than multigrid \cite{bcdl}  or
multipole methods \cite{lemou} but this drawback can be compensated by a better
accuracy at least asymptotically when $N\gg 1$.   Let us emphasize
that more recently, spectral algorithms based on Hermite expansion of the
distribution function have been proposed \cite{LRW} allowing
to get exact conservations of mass, momentum and energy but the
computational cost is again larger than $N\,\log_2N$. This latter
method is particularly efficient when the solution remains closed to
the equilibrium since few modes may be used.

In the present paper, we pursue the idea of spectral methods, but in
the frame of spectral collocation methods, where the distribution
function is now discretized on a grid of the velocity space instead of
computing the time evolution of Fourier modes. Fast Fourier transforms
are only applied to evaluate non-local convolution terms and discrete
derivatives. Here we will follow the idea of the spectral method
described in \cite{PRT1, PRT2} and restrict ourselves to the Coulombian case
$\gamma=-3$, which has a particular structure. We will take
advantage of this framework  to reduce the number of discrete convolutions.

The rest of the article is organized as follows. In the next section
we describe the main features of our numerical methods and propose the
a new spectral collocation method for the approximation
of the collision operator. Next we discuss some properties of the
numerical scheme as spectral accuracy and preservation of steady states. Finally, several numerical tests for three dimensional
space homogeneous problems are presented to illustrate the efficiency
of the spectral collocation method.

\section{The numerical method}
\label{sec:nm}
\setcounter{equation}{0}
From now, we only consider the Coulombian case ($\gamma=-3$), which is
the most significant for a physical point of view. In that case, the
Landau operator has an additional property which will be the key point
of our approximation. We set $\psi(\bz)=|\bz|$ and observe that the
matrix $\bA$ in \eqref{eq:1} is such that
$$
\bA(\bz) \,=\,  \nabla^2 \psi (\bz).
$$
Hence, the Landau operator \eqref{eq:1} can be now written as
\be
\left\{
  \begin{array}{l}
    g = \psi\star f\,,
    \\[0.9em]
\cC(f,f) = \Div_\bv\left(\nabla_\bv^2 g\,
    \nabla_\bv f \,-\, \nabla_\bv\Delta_\bv g \,f\right).
    \end{array}\right.
\label{eq:3}
\ee
This formulation will be the key point of our spectral collocation
method. Unfortunately, this formulation is not the most appropriate to
prove conservation of momentum and energy. Indeed, to prove conservation of
momentum, we may use that $g$ satisfies
$$
-\Delta^2 g \,=\, 8\,\pi \, f \quad{\rm in}\,\R^3,
$$
whereas the conservation of energy requires that we come back to the
original formulation and observe that $\bz \in \ker\bA(\bz)$. However,
this formulation is convenient  to reduce the
computational cost since  the scalar and non-local function $g$
may be evaluated using a discrete fast Fourier transform.  Moreover, once the function $g$ is provided, the operator $\cC(f,f)$
is a local convection/diffusion operator, hence it can be approximated
either by finite difference formula or spectral approximation.  Let us
emphasize that this formulation is widely used in physics where the potential $g=\psi\star f$ refers to the Rosenbluth potential \cite{RMJ}.

Now let us explain our spectral collocation method for \eqref{eq:3}. We  consider a set of equidistant points $(\bv_\bj)_{\bj\in \J_{n}} \subset [-R,R]^3$ with  $\J_{n}:=\llbracket
-n/2 ,n/2-1\rrbracket^3$ where $n$ is an even integer and a real
$R>0$, which determines the computational domain in velocity.   Let us
suppose that  an approximation of the distribution function is known at the mesh points $(\bv_\bj)_{\bj\in
  \J_{n}}$.

\subsection{Computation of the convolution term}
Similarly to  classical spectral methods for Boltzmann or Landau equations
\cite{PP, PRT2, FMP},  the first step will consists to reduce the
integration domain $\R^3$ to a bounded domain.  It can be shown that for a collision operator such
as  \eqref{eq:3}, we have the following property
\bpr
  \label{prop:1}
  Let $\Supp ( f ) \subset \cB(0, R)$, where $\cB(0, R)$ is the ball of radius $R$ centered in the origin. Then
  \be
  \label{eq:C}
\left\{
  \begin{array}{l}
    g \,=\, \psi^R\star f\,,
    \\[0.9em]
\cC(f,f) = \Div_\bv\left(\nabla_\bv^2 g\,
    \nabla_\bv f \,-\, \nabla_\bv\Delta_\bv g \,f\right)\,,
  \end{array}\right.
\ee
with $\psi^R = \chi_{[-2R,2R]^3}\,\psi$, where $\chi_B$ denotes the
characteristic function in the set $B$. Moreover, we have $\bv-\bu\in \cB(0,3\,R)$.
\epr
\begin{proof}
First we  write  $\cC(f,f)$ as
  $$
\cC(f,f)(\bv) \,=\, \Div_\bv\left(\nabla_\bv^2\int_{\R^3}\psi(\bu)
  f(\bv-\bu) \dD \bu\,
  \nabla_\bv f(\bv) \,-\, \nabla_\bv\Delta_\bv \int_{\R^3}\psi(\bu)
  f(\bv-\bu)\dD\bu\,f(\bv)\right)
  $$
  and consider  that $f$ is compactly supported in a ball $\cB(0,R)$. On the one hand,  for  $\bv$ and $\bv - \bu \in \cB(0, R)$,
  $$
|\bu| \,=\, |\bv - \bv - \bu| \,\leq\, |\bv| \,+\, |\bv + \bu| \,\leq\,  2\,R,
$$
hence in this case, it is  enough to choose $\bu\in [-2R,2R]^3$. On the other hand, when  $\bv$ or $\bv -\bu \notin \cB(0, R)$ we get
that  $\cC(f, f)=0$ since $f\equiv 0$ in $\R^3\setminus \cB(0,R)$.

Therefore, the Landau operator may be written as
$$
 \cC(f,f) \,=\, \Div_\bv\left(\nabla_\bv^2\left[\psi^R\star f\right]\,
  \nabla_\bv f \,-\, \nabla_\bv\Delta_\bv \left[\psi^R\star
    f\right]\,f\right),
$$
which gives formula \eqref{eq:C}.

Finally $|\bv| \leq R$ and $|\bu| \leq 2R$ implies
$$
|\bv-\bu| \leq |\bv| + |\bu| \leq 3R.
$$
\end{proof}

Following Proposition \ref{prop:1}, we want to discretize the
Landau equation in a computational domain $[-R,R]^3$, with $R>0$,
hence we define a truncated collision operator $\cC^R(f,f)$ as
\be
  \label{eq:CR}
\left\{
  \begin{array}{l}
    g^R = \psi^R\star f\,,
    \\[0.9em]
\cC^R(f,f) = \Div_\bv\left(\nabla_\bv^2 g^R\,
    \nabla_\bv f \,-\, \nabla_\bv\Delta_\bv g^R \,f\right).
  \end{array}\right.
\ee

\begin{remark}
Let us emphasize that this truncation of the domain will affect in
practice the exact conservation of momentum and energy. We refer to
\cite{bobylev0, degond} or
more recently  \cite{chacon1, chacon2} for specific numerical schemes ensuring these
conservations by modifying boundary terms. Here we favor on spectral
accuracy instead of exact conservations.
\end{remark}

As for spectral approximations to Boltzmann \cite{PP,PR}
and Landau \cite{PRT2,FP}, we aim to compute the convolution term
$g^R$ using a discrete fast  Fourier transform. To avoid aliasing
effects \cite{CHQZ},  we
need to extend the distribution function  $f$ carefully in a larger domain. Following
\cite{PP,PR, FMP} and Figure \ref{fig:0}, we choose  the cube $[-T ,
T ]^3$, where  $f\,=\, 0$ on $[-T, T ]^3 \setminus [-R,R]^3$, and extend it by periodicity to a
periodic function on $[-T , T ]$. As observed in \cite{PP,PRT2} and
illustrated in Figure \ref{fig:0}, it is enough to take $T
\geq 3\,R/2$ to prevent intersections of the regions where $f$ is different from
zero. In practice we take $T=2\,R$ and consider a set of equidistant points
$(\bv_\bj)_{\bj\in \J_{2n}} \subset [-T,T]^3$ with $T=2R$ with $\J_{2n}:=\llbracket
-n ,n-1\rrbracket^3$.

\begin{figure}
\newcommand{\ttE}{(-3,-3) rectangle (3,3)}
\newcommand{\ttF}{(3,-3) rectangle (9,3)}
\newcommand{\ttA}{(0,0)  circle (2)}
\newcommand{\ttB}{(6,0)  circle (2)}
\newcommand{\ttC}{(0,0)  circle (4)}
\newcommand{\ttD}{(6,0)  circle (4)}
\begin{center}
\begin{tikzpicture}
\draw [<->,>=stealth',line width=1pt] (9.5,0) -- (0,0) -- (0,4.5);
\draw[color=red] \ttE;
\draw[color=red] \ttF;
\fill[opacity=0.5,red]  \ttA;
\fill[opacity=0.5,red] \ttB;
\draw[black, dashed] \ttC;
\draw[black, dashed] \ttD;
\draw (0,0) node{$\bullet$};
\draw (0,0) node[below left]{$0$};
\draw (3,0) node{$\bullet$};
\draw (3,0) node[below left]{$T$};
\draw (6,0) node{$\bullet$};
\draw (6,0) node[below left]{$2T$};
\draw (9,0) node{$\bullet$};
\draw (9,0) node[below left]{$3T$};
\draw (0,-1.5) node{$\cB(0,R)$};
\draw (-0.5,-3.5) node{$\cB(0,2R)$};
\end{tikzpicture}
\end{center}
\caption{Computation of the cube $[-T,T]^3$ to avoid interactions
  between the  domain of integration $\cB(0,2R)$ and the periodized
  distribution function $f_n$ for the convolution term
  $g_{2n}^R=\psi^R\star f_{2n}$. A sufficient condition is $T>0$ is such that  $2T\geq 3\,R$.}
\label{fig:0}
\end{figure}

We suppose that  $f$ is only known at the mesh points $(\bv_\bj)_{\bj\in
  \J_{n}}\subset [-R,R]^3$ and set $f(\bv_\bj)=0$ for $\bj\in
\J_{2n}\setminus \J_n$.  Then we compute a discrete  Fourier transform as,
$$
\widetilde{f}_{2n}(\bk) \,:=\, \dfrac{1}{(2n)^3}\,\ds\sum_{\bj\in \J_{2n}}
f(\bv_\bj)\,e^{-\frac{i \pi}{T}\,\bk\cdot\bv_\bj}, \quad \bk\in\J_{2n}
$$
and get a function  $f_{2n}$ as a trigonometric polynomial  in the domain $[-T,T]^3$,
$$
f_{2n}(\bv)\,:=\,\ds\underset{\bk\in
  \J_{2n}}{\sum}\,\widetilde{f}_{2n}(\bk)\,e^{\frac{i\,\pi}{T}\,\bk\cdot\bv}.
$$
Due to the orthogonality relation
\begin{equation}
\frac{1}{(2n)^3} \sum_{\bj\in\J_{2n}}
e^{-\frac{i\,\pi}{T}\,\bk\cdot\bv_j} \,=\,
\left\{\begin{array}{ll}
         1, & {\rm if\,} \bk=\, 2n \,m\, {\bf 1}, \quad m=0,\,\pm 1,\,\pm 2,\ldots
         \\
         0,  &{\rm else},
       \end{array}\right.
\label{p:ortho}
     \end{equation}
     where ${\bf 1}=(1,1,1)$, we have the relation for any $\bj\in\J_{2n}$,
     $$
f_{2n}(\bv_\bj)\,=\,\ds\underset{\bk\in
  \J_{2n}}{\sum}\,\widetilde{f}(\bk)\,e^{\frac{i\,\pi}{T}\,\bk\cdot\bv_\bj}
\,=\, f(\bv_\bj).
     $$
Consequently $f_{2n}$ is the $n$-degree trigonometric interpolant of
$f$ at the nodes $(\bv_\bj)_{\bj\in \J_{2n}}$, that is,
$f_{2n}(\bv_\bj)\,=\, f(\bv_\bj)$ for all $\bj\in\J_{2n}$.

The first step consists in computing the convolution term
$g_{2n}^R$ by substituting the function $f_{2n}$ in the first equation of \eqref{eq:CR},  which yields
 \begin{equation}
  \label{def:gn}
g_{2n}^R \,:=\, \sum_{\bk\in \J_{2n}} \widetilde{g}_{2n}(\bk)\, e^{\frac{i\,\pi}{T}\,\bk\cdot\bv},
\end{equation}
where  $\widetilde{g}_{2n}$ is simply given by
\begin{equation}
  \label{def:hatL0}
  \widetilde{g}_{2n} (\bk) \,\,=\,\, \widetilde f _{2n} (\bk) \, \widetilde{\psi}(\bk),
\end{equation}
with
$$
  \widetilde{\psi}(\bk) \,=\,  \int_{[-T,T]^3}
  \psi^R(|\bu|)\,e^{-\frac{i\,\pi}{T}\,\bk\cdot\bu}\,\dD\bu \,=\,  \int_{[-T,T]^3}
  |\bu|\,e^{-\frac{i\,\pi}{T}\,\bk\cdot\bu}\,\dD\bu.
  $$
  This kernel  depends on the computational domain of  $f$ through the
  period $T$, hence in order to make this dependence explicit, we
  apply a simple change of variable and get
  \begin{equation}
  \label{def:hatL1}
\widetilde{\psi}(\bk) \,=\,  \left(\frac{T}{\pi}\right)^4\,\int_{[-\pi,\pi]^3}
  |\bz|\,e^{-i\,\bk\cdot\bz}\,\dD\bz.
  \end{equation}
  Therefore, the computation of $\widetilde{\psi}(\bk) $ can be
  performed easily  by computing the Fourier coefficients corresponding to the periodic
  function $\bu \mapsto |\bu|$.

\subsection{Computation of $\cC(f_n,f_n)$}
From the trigonometric polynomials $g_{2n}^R$ defined in the cube
$[-T,T]^3$, we want to evaluate the collision operator $\cC$ at the
collocation points $(\bv_\bj)_{\bj\in\J_n}\,\subset \,[-R,R]^3$.
Therefore, we compute a new approximation $f_{n}$ as a trigonometric polynomial  in the domain $[-R,R]^3$,
$$
f_{n}(\bv)\,:=\,\ds\underset{\bk\in
  \J_{n}}{\sum}\,\widetilde{f}_{n}(\bk)\,e^{\frac{i\,\pi}{R}\,\bk\cdot\bv}\,,
$$
with
$$
\widetilde{f}_{n}(\bk) \,:=\, \dfrac{1}{n^3}\,\ds\sum_{\bj\in \J_{n}}
f(\bv_\bj)\,e^{-\frac{i \pi}{R}\,\bk\cdot\bv_\bj}, \quad \bk\in\J_{n}\,,
$$
such that  $f_{n}(\bv_\bj)\,=\, f(\bv_\bj)$ for all $\bj\in\J_{n}$.

Therefore, we define the interpolation operator $\proj_{n}$ such that
$f_{n}=\proj_{n}f$, which can be considered as the orthogonal
projection upon the space  $\cP_{n}$ of trigonometric polynomials of
degree $n$, with respect to the discrete approximation of the $L^2$
inner product,
$$
\cP_{n} \,:=\, {\rm span}\left\{ e^{\frac{i \pi}{R}\,\bk\cdot\bv},
  \quad \bk=(k_1,k_2,k_3),\, {\rm \,with\,} k_\alpha=-n/2,\ldots,n/2-1,
  \,{\rm \,for\,}\alpha\in\{1,\,2,\,3\}\right\}.
$$
Actually , the bilinear form
$$
\langle f, h \rangle  \,=\, \frac{1}{n^3} \sum_{\bj\in\J_{n}}  f(\bv_j)
\, h(\bv_j)
$$
coincides with the inner product of $L^2([-R,R]^3)$ when $f$ and $h \in
\cP_{n}$.

The interpolant $\proj_{n}f$ of a continuous function $f$ satisfies
the identity
$$
\langle \proj_{n}f, h \rangle  \,=\, \langle f, h \rangle, \qquad h\in\cP_{n}.
$$
Thus, from $f_{n}$ and $g_{2n}$, we compute an approximation $\cC_n^R(f_{n},f_{n})$ as
\begin{eqnarray*}
\cC_n^R(f_{n},f_{n}) &=& \Div_\bv \proj_{n}\left(\nabla_\bv^2 g_{2n}^R\,
    \nabla_\bv f_{n} \,-\, \nabla_\bv\Delta_\bv g_{2n}^R
                     \,f_{n}\right),
  \end{eqnarray*}
Now for the Fourier collocation method, we will require that the residual
$$
\cR_n = \frac{\partial f_n}{\partial t} -  \cC_n^R(f_n,f_n)
$$
vanishes at the grid points $(\bv_\bj)_{\bj\in\J_{n}}\subset [-R,R]^3$, that is,
$$
\cR_{n}(t,\bv_j) \,=\,0,\quad \bj \in\J_{n}.
$$
This yields $n^3$ equations to determine the point values
$f_n(t,\bv_j)$ of the numerical solution.  In other words, the
pseudospectral approximation $f_n$ satisfies the equation
\be
\label{eq:24}
\frac{\partial f_n}{\partial t} \,\,=\;\;  \cC_n^R(f_n,f_n).
\ee
Notice that from $g_{2n}^R$ defined in $[-T,T]^3$, we only compute $\nabla_\bv^2 g_{2n}^R$ and
$\nabla_\bv\Delta_\bv g_{2n}^R$  at the collocation points $(\bv_\bj)_{\bj\in\J_{n}}$ in the smaller cube $[-R,R]^3$ to evaluate
$\cC_n^R(f_n,f_n)$.

\section{Properties of the pseudospectral method}
We point out that because of the periodicity  assumption on the
operator $\proj_n$, the collision operator $\cC^R_n( f_n,f_n )$ preserves in time
the mass
$$
\int_{[-R,R]^3} \cC^R_n( f_n,f_n ) \dD \bv \,=\, \ \int_{[-R,R]^3}
\Div_{\bv}\proj_n(\nabla_\bv^2 g_{2n}^R \nabla_\bv f_n -
\nabla_\bv\Delta_\bv f_n ) \dD \bv \,=\, 0.
$$
In contrast, momentum and energy are
not exactly preserved in time but as we will see below these
variations are controlled by spectral accuracy when the solution is smooth.

\subsection{Spectral accuracy}
As for spectral methods \cite{PRT2}, we can
state the following theorem.

\begin{theorem}
  \label{th:1}
  Consider $\psi^R$ given by \eqref{eq:C} and  a distribution function $f \in H^p$, with $p\geq 4$, which
  is compactly supported in the ball $\cB(0,R)$ with $R>0$. Then there
  exists a nonnegative constant $C^R>0$,  depending on  $R$, such that
  $$
\| \cC^R(f,f) - \cC^R_n(f_n,f_n) \|_{L^2}
 \,\leq\,\frac{C_R}{n^{p-4}} \, \|f\|_{H^p}\,\|f\|_{H^4}.
  $$
\end{theorem}
\begin{proof}
 First for  $f \in H^{r}$, with $r\geq 1$,  we consider $g^R=\psi^R\star f$, where $\psi^R$ is
 continuous and $\psi^R\in L^1\cap L^\infty$, hence we  get from the
Young's convolution inequality, for any $r\geq 0$
\begin{equation}
  \label{esti:0}
  \left\{
    \begin{array}{l}
\ds\|\nabla_\bv^r g^R\|_{L^\infty} \leq \|\psi^R\|_{L^2}\, \|\nabla_\bv^r
      f\|_{L^2}\,,
      \\[0.9em]
      \ds\|\nabla_\bv^r g^R\|_{L^2} \leq \|\psi^R\|_{L^1}\, \|\nabla_\bv^r
      f\|_{L^2}\,.
      \end{array}\right.
\end{equation}
Also let us  remind the following result in approximation theory : for $f\in
H^{l_0}_\#$, we have for any $0\leq l \leq l_0$,
\begin{equation}
\label{esti:sp}
\| f - \proj_n f \|_{H^l} \,\leq \, C_R \,
\frac{\|f\|_{H^{l_0}}}{n^{l_0-l}}.
\end{equation}
We now introduce
$$
\left\{
  \begin{array}{l}
\cB^R_1(f,f) =  \Div_\bv \proj_n\left(\nabla_\bv^2 g^R\,
    \nabla_\bv f \,-\, \nabla_\bv\Delta_\bv g^R
                     \,f\right),
    \\[0.9em]
   \cB^R_2(f,f) =  \Div_\bv \proj_n\left(\nabla_\bv^2 g_{2n}^R\,
    \nabla_\bv f \,-\, \nabla_\bv\Delta_\bv g_{2n}^R
    \,f\right),
    \end{array}\right.
    $$
    and estimate the numerical error as
    \begin{eqnarray}
      \label{t:0}
\| \cC^R(f,f)  - \cC_n^R(f_n,f_n) \|_{L^2} &\leq&  \| \cC^R(f,f)  -
                                                  \cB_1^R(f,f) \|_{L^2}
      \\
      &+& \| \cB^R_1(f,f)  - \cB_2^R(f,f) \|_{L^2}
\nonumber
      \\    &+& \| \cB_2^R(f,f)  - \cC_n^R(f_n,f_n) \|_{L^2}.
          \nonumber
\end{eqnarray}
The first term on the right hand side of \eqref{t:0} measures the interpolation error
of trigonometric polynomials. We set $h\,=\,\nabla_\bv^2g^R\,\nabla_\bv f - \nabla_\bv\Delta_\bv g^R
\,f$ and observe that since $f$ is compactly supported in $\cB(0,R)$,
the function $h$ is also compactly supported, hence it is periodic
and belongs tp $H^{p-3}_{\#}([-R,R]^3)$. Indeed, from the H\"older
inequality and next  the first Young convolution's inequality \eqref{esti:0}, we get
\begin{eqnarray*}
  \| h \|_{H^{p-3}}
&\leq& C\,\left( \|g^R\|_{W^{p-1,\infty}} \,
                         \| f\|_{H^{p-2}} \,+\,  \|g^R\|_{W^{p,\infty}} \,
                         \| f\|_{H^{p-3}} \right)\,,  \\
  &\leq& \, C\,\|g^R\|_{W^{p,\infty}} \,
                         \| f\|_{H^{p-2}}\,, \\
  &\leq&   C\,\|\ \psi^R\|_{L^2} \, \| f\|_{H^{p}}\,\| f\|_{H^{p-2}}\,.
\end{eqnarray*}
Applying \eqref{esti:sp} to $h$ with $l=1$ and $l_0=p-3$, it yields
$$
\| \cC^R(f,f)  -\cB_1^R(f,f) \|_{L^2}  \leq  \|\proj_nh - h \|_{H^1}
\leq C_R \,\frac{\|h\|_{H^{p-3}}}{n^{p-4}}\,,
$$
hence,
\begin{equation}
\label{t:1}
\| \cC^R(f,f)  -\cB_1^R(f,f) \|_{L^2}  \leq  \frac{C_R}{n^{p-4}} \,  \|\psi^R\|_{L^2}\, \|f\|_{H^p} \, \|f\|_{H^{p-2}}\,.
\end{equation}
The second term in \eqref{esti:0} corresponds to the error between
$g^R$ and its trigonometric interpolant $g_{2n}^R$ defined in the cube
$[-T,T]^3$. Let us notice that since $g_{2n}^R \in
\cP_{2n}([-T,T]^3)$ with $T=2R$, whereas $f$ is compactly supported in
$\cB(0,R)$, as a consequence the function $\nabla_\bv^2 g_{2n}^R \,\nabla_\bv f -
\nabla_\bv\Delta_\bv g_{2n}^R \, f$ is smooth and compactly supported in
$\cB(0,R)$ and its restriction to the cube  $[-R,R]^3$ can be regarded
as a smooth and periodic function. Again applying the
stability estimate to any $u\in H^1_\#$
$$
\|\Div_\bv \left(\proj_n u\right)\|_{L^2}\,\leq \,\|\proj_n u\|_{H^1}\,\leq\, C\, \|u\|_{H^1}
$$
and the H\"older inequality, we get
\begin{eqnarray*}
\| \cB_1^R(f,f)   -\cB_2^R(f,f) \|_{L^2}  & \leq & \|\nabla_\bv f \,
\nabla_\bv^2 (g^R - g^R_{2n}\|_{H^1} \,+\, \| f \, \nabla_\bv\Delta_\bv (g^R - g_{2n}^R)\|_{H^1}\,,
  \\
                                          & \leq & C\,\left(
                                                   \|\nabla_\bv^2
                                                   f\|_{L^\infty}
                                                   \,+\, \|f\|_{L^\infty} \right) \, \| g^R -  g^R_{2n} \|_{H^4}.
\end{eqnarray*}
 Thus, using that $f$ and $\nabla_\bv^2 f \in H^2 \subset L^\infty$
 with continuous embedding, then  applying \eqref{esti:sp}  to $g^R$ with $l=4$ and $l_0=p$ and next \eqref{esti:0}, it
 gives the following estimate on the second term of the right hand
 side in \eqref{t:0},
\begin{equation}
\label{t:2}
  \| \cB_1^R(f,f)   -\cB_2^R(f,f) \|_{L^2}  \,\leq\,\frac{C_R}{n^{p-4}} \,  \|\psi^R\|_{L^1}\, \|f\|_{H^p}\, \|f\|_{H^4}.
\end{equation}
Finally we estimate the third term in \eqref{t:0} which takes into account
the error between $f$ and its interpolant $f_n$ in the cube
$[-R,R]^3$. We have already seen that $\cB_2^R(f,f)$ is compactly
supported and it can be considered
as a periodic smooth function in $[-R,R]^3$ whereas $\cC_n^R$ is
the divergence of the interpolant of the  function $h_n\,=\,\proj_n\left(\nabla_\bv^2
g^R_{2n} \nabla_\bv f_n - \nabla_{\bv}\Delta_\bv g_{2n}^R\,
f_n\right)$, which is also periodic in $[-R,R]^3$ and infinitely smooth.
\begin{eqnarray*}
\| \cB_2^R(f,f) - \cC^R_n(f_n,f_n)  \|_{L^2}  &\leq& \left( \|\nabla_\bv^2
g_{2n}^R\|_{L^\infty}+ \|\nabla_\bv^3
                                                     g_{2n}^R\|_{L^\infty}\right) \, \| \nabla_\bv( f- f_n)\|_{H^1}
  \\
  &+&
\left( \|\nabla_\bv^3 g_{2n}^R\|_{L^\infty}+\|\nabla_\bv^4 g_{2n}^R\|_{L^\infty}\right) \, \|  f- f_n\|_{H^1},
\end{eqnarray*}
which gives applying the same argument as before,
\begin{equation}
\label{t:3}
  \| \cB_2^R(f,f) - \cC^R_n(f_n,f_n) \|_{L^2}  \,\leq\,\frac{C_R}{n^{p-2}} \,  \|f\|_{H^p}\, \|\psi^R\|_{L^2}\, \|f\|_{H^4}.
\end{equation}
Gathering \eqref{t:1}-\eqref{t:3} into \eqref{t:0}, we get the spectral
accuracy, there exists a constant $C_R>0$ such that,
$$
 \| \cC^R(f,f) - \cC^R_n(f_n,f_n) \|_{L^2}
 \,\leq\,\frac{C_R}{n^{p-4}} \, \|f\|_{H^p}\,\|f\|_{H^4}.
$$

\end{proof}

\begin{remark}
  Let us emphasize that in \cite[Corollary 2.1]{PRT1, PRT2}, the
  consistency error is slightly  better since with the
  same assumptions on $f\in H^p$, the authors get
  $$
\| \cC^R(f,f) - \cC^R(f_n,f_n) \|_{L^2}
 \,\leq\,\frac{C_R}{n^{p-2}} \, \left(\|f\|_{H^p}\,+\, \|\cC^R(f_n,f_n)\|_{H^p}\right),
 $$
 where $\cC^R(f_n,f_n)$ represents now a spectral approximation. In
 our case, we provide an estimate of the last term in the right-hand
 side for the Coulombian case, hence  the error is deteriorated.
  \end{remark}
No information is available on the discrete equilibrium states,
the decay of the numerical entropy and the preservation of positivity.

\subsection{Comparison with the classical spectral method \cite{PRT2}}

We have seen in the previous section that the spectral collocation
method provides an approximation of the collision operator with
spectral accuracy and preserves mass as the classical spectral method does
\cite{PRT2}. However, in term of computational efficiency the spectral
collocation method  has some advantages.

On the one hand,  for a distribution function $f$ such that $\Supp f
\subset \cB(0,R)$, the computational domain is kept relatively small
$[-R,R]^3$,   since the distribution function is simply extended to zero for the
  evaluation of the convolution term $g=\psi\star f$, whereas the
  computational domain of the spectral method is taken as $[-T,T]^3$
  with $T\simeq 2R$.

On the other hand,  as it is pointed out in \cite[Section 3]{PRT2}, the
  evaluation of the Fourier kernel of $\cC(f,f)$ requires nine
  convolutions of the form
  $$
  \widehat{q}(\bk) \,=\, \sum_{\bll+\bmm=\bk} \alpha(\bll)\,\widehat{f}(\bll)\,  \beta(\bmm)\,\widehat{f}(\bmm)\,,
  $$
where $\widehat{f}(\bll)$ is the Fourier coefficient of $f$. Each
convolution term  requires three discrete Fourier transforms  with a
computational cost of $O\left((2n)^3\,\log((2n^3))\right)$ to avoid
aliasing \cite{CHQZ}, where $n^3$ represents the total number of
Fourier coefficients. This computational cost is mainly due to the
fact that $\cC(f,f)$ is quadratic with respect to $f$, hence
nonlinear terms become  discrete convolutions in Fourier variable,
whereas the non-local term becomes local in Fourier variable.

Here we take advantage of the Coulombian case and the fact we
discretize the nonlinear operator in the velocity space. Indeed, first the matrix
  \eqref{def:A} can be written as $\nabla^2_\bv \psi$, hence we only
  compute one convolution term $g=\psi\star f$, which corresponds to a
  nonlocal term in velocity then we differentiate
  its trigonometric  polynomial approximation.  The cost of this
  convolution is reduced since $\psi$ is fixed and its discrete
  Fourier coefficient might be stored.   Finally, since we compute
  $\cC(f,f)$ at collocation points $(\bv_\bj)_{\bj\in\J_n}\subset
  [-R,R]^3$, the nonlinear terms do not require an additional cost,
  whereas the non-local term is a convolution, which can be evaluated
  with a cost $O((2n)^3\log((2n)^3))$ to compute $f_{2n}$.

  Finally the last comment concerns the Fourier coefficients. The
  classical spectral method may be written in the form \cite[Section 3]{PRT2},
  $$
\frac{\partial \widehat{f}(\bk)}{\partial t} \,=\, \sum_{\bmm\in
\J_{n}} \widehat{\beta}(\bk-\bmm,\bmm)  \widehat{f}(\bmm)\, \widehat{f}(\bk-\bmm)\,,
$$
with
$$
\widehat{\beta}(\bll,\bmm) = |\bll |^2 \, {\rm
  Tr}\left(\cI(\bmm)\right) \,-\, \bll^t \,\cI(\bmm)\,\bll\,,
$$
and
$$
\cI(\bmm) \,=\, \int_{\cB(0,\pi)} \frac{1}{|\bz|} \,\left( \bI - \frac{\bz\otimes \bz}{|\bz|^2}\right)\, e^{-i\bmm\cdot \bz}\dD \bz\,.
$$
This latter integral can be reduced to a one dimensional integral but
the integrand has a singularity in $\bz=0$, hence it has to be
computed carefully using a  recursive quadrature formula. With the
spectral collocation, we only need to evaluate \eqref{def:hatL1},
given by
$$
\widetilde{\psi}(\bk) \,=\,  \left(\frac{T}{\pi}\right)^4\,\int_{[-\pi,\pi]^3}
  |\bz|\,e^{-i\,\bk\cdot\bz}\,\dD\bz,
  $$
  which can be done quickly using a fast Fourier transform to $\bu
  \mapsto |\bu|$. This latter function is not infinitively smooth in the
  cube $[-\pi,\pi]^3$, hence a large number of collocation points  has
  to be used to evaluate $\widetilde{\psi}(\bk)$ and to preserve
  spectral accuracy. Let us notice that in the proof of Theorem \ref{th:1}, we did not
  tale into account the error due to the approximation of these
  Fourier coefficients.

  \subsection{Steady-state-preserving method}
  A major drawback of \eqref{eq:24} is the lack of exact conservations and,
  as a consequence, the incapacity of the scheme to preserve
  Maxwellian distribution function. In this section, we overcome this
  drawback thanks to a new reformulation of the method already propose
  in the context of Boltzmann equation \cite{FPR} that permits to
  preserve the spectral accuracy and to capture the long-time behavior
  of the system.

Let us denote  $\cM_n$  a trigonometric polynomial belonging to
$\cP_n$ and
interpolating the Maxwellian distribution $\cM$, given in \eqref{eq:M}, at the collocation
points $(\bv_\bj)_{\bj\in\J_n}$. Note that due to space homogeneity,
$\cM$ does not change in time and so does $\cM_n$. We simply modify the
previous method \eqref{eq:24} into
\be
\label{eq:24-bis}
\frac{\partial f_n}{\partial t} \,\,=\;\; \cL^R_n(f_n,f_n)\,,
\ee
where
$$
\cL^R_n(f_n,f_n)\,:=\,  \cC_n^R(f_n,f_n) - \cC_n^R(\cM_n,\cM_n)\,.
$$
Observe that thanks to Theorem \ref{th:1}, this additional term does
not affect the spectral accuracy for smooth solutions to the Landau
equation \eqref{eq:0}-\eqref{eq:1}. Indeed, we have
$$
\| \cC^R_n(\cM_n,\cM_n) \|_{L^2}\,=\,\| \cC^R_n(\cM_n,\cM_n)-\cC^R(\cM,\cM) \|_{L^2}
 \,\leq\,\frac{C_R}{n^{p-4}} \, \|\cM\|_{H^p}\,\|\cM\|_{H^4}.
 $$
 Hence we get the same result as Theorem \ref{th:1} to
 \eqref{eq:24-bis} in term of accuracy but also the conservation of the
 consistent steady state $\cM_n$. We will illustrate in the next
 section the advantage of the steady state preserving method for the
 long time behavior of the solution to \eqref{eq:0}-\eqref{eq:1}.

  \section{Numerical simulations}

In this section we perform some numerical tests of the scheme
\eqref{eq:24} and \eqref{eq:24-bis}, to
check the accuracy and the efficiency of the method.

All calculations have been performed by a third-order Runge-Kutta
scheme, with fixed time step.  Of course, the Landau equation suffers
from the stiffness typical of diffusion equations. The stability
condition requires that the time step scales with the square of the
velocity step. This means that by doubling the number of Fourier modes
per direction, the total number of time steps becomes four times
bigger to compute up to the same final time. We have not performed a
stability analysis of the scheme, and the stability condition used in
the computation has been found empirically. No attempt has been made
to overcome the numerical stiffness of the problem caused by
diffusion. Although this is a very important issue and deserves a
careful study, it is beyond the scope of the present paper and we
refer to \cite{LM05} or \cite{FJ, BFR} for research
directions on this topic. In our numerical test, we first look for the
time step $\Delta t$ when $n$ is small and then choose $\Delta
t=O(1/n^2)$ when $n$ is increased.

\subsection{Order of accuracy}
Unfortunately for Coulombian interactions, there is no explicit
solution to the evolution problem \eqref{eq:0}  as it holds true when
$\gamma=0$ in \eqref{eq:1}-\eqref{def:A}. However, since the
Maxwellian distribution \eqref{eq:M} is a stationary state for
\eqref{eq:1}-\eqref{def:A} and since the spectral collocation method
\eqref{eq:24} does not preserve  steady states, the Maxwellian may be
used to evaluate the accuracy. To this aim we choose the initial data
$$
f_0 = \cM_{1, 0,1}  \quad {\rm in }\, [-7,7]^3,
$$
with a small time step $\Delta t=0.005$. Then, we perform numerical
simulations on the time intervall $t \in [0,1]$ with respectively
$n^3=8^3$, $n^3=16^3$ and $n^3=32^3$. In Figure \ref{fig:t1-1}, we
represent the time evolution of the the relative entropy
\eqref{h:rel} and also  $L^1$, $L^2$ and $L^\infty$
error norms. The rate at which the error decays with the increase of the
number of modes is an indication of spectral accuracy. Let us
emphasize that as we mentioned before, the crucial point is to get an
accurate approximation of Fourier coefficients $(\widetilde{\psi}(\bk)
)_\bk$ in order to get spectral accuracy. Indeed, for this test we use
$1024$ collocation points in each direction to evaluate
\eqref{def:hatL1}.

\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
    \includegraphics[width=8cm]{t1-f2.png} &
  \includegraphics[width=8cm]{t1-f3.png}
      \\
 (a) $\|f_N-\cM_{1,0,1}\|_{L^2}$ & (b) $\|f_N-\cM_{1,0,1}\|_{L^\infty}$
\end{tabular}
  \end{center}
\caption{{\bf Order of accuracy:} time evolution of  the $L^2$ and $L^\infty$ error norms for
  the scheme \eqref{eq:24}.}
\label{fig:t1-1}
\end{figure}

\begin{table}[!ht]
	\centering
	\vspace{0.15in}
	\begin{tabular}{|l|l|l|l|l|}
          \hline
           $n$ & $L^2$ error norm & Order &
                                                                    $L^\infty$ error norm & Order \\
          \hline
          08 &  1.66\,$\times \,10^{-2}$ & --   &  7.03\, $\times \,10^{-3}$  & --
          \\ \hline
          16 & 5.73\,$\times \,10^{-5}$ & 8.17 & 2.30\,$\times \,10^{-5}$ & 8.24\\
          \hline
          32 & 4.46\,$\times \,10^{-9}$ & 13.64 & 3.99\,$\times \,10^{-9}$ & 12.9  \\
          \hline
        \end{tabular}
	\caption{{\bf Order of accuracy:} numerical error for $L^2$
          and $L^\infty$ norms and order of accuracy.}
	\label{tab:1}
      \end{table}

\subsection{Rosenbluth problem}
We choose the initial condition as
$$
f_0(\bv) \,=\,
\frac{1}{S^2} \exp\left( -S\,\frac{(|\bv| - \sigma)^2}{\sigma^2}\right),
$$
 with $\sigma = 0.3$ and $S = 10$ and the integration time is $T_0
 =50$ with $\Delta t = 0.1$ in the computational domain $[-1,1]^3$.

 This test is used to compute the time evolution of the numerical solution \eqref{eq:24-bis} and to compare the
 results with those obtained in \cite{RMJ, PRT2}.

 First, we again illustrate the spectral accuracy by plotting the time
 evolution of total momentum and temperature which are not exactly
 preserved by the spectral collocation method. In  Figure
 \ref{fig:t2-1}, we present the time evolution of momentum and
 temperature in log scale and observe that their variations become
 smaller and smaller when the number of collocation point is
 increasing. With only $n=24$, the variations of momentum and
 temperature are of order $10^{-5}$.
\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[width=8cm]{t2-f5.png} &
  \includegraphics[width=8cm]{t2-f6.png}
      \\
 (a) $\|\rho\,\bu(t)\|$ & (b) $|T(t)-T(0)|$
\end{tabular}
  \end{center}
\caption{{\bf Rosenbluth problem:} time evolution of the variations of
  momentum \eqref{def:rho-u} and
  temperature \eqref{def:T}}
\label{fig:t2-0}
\end{figure}

We compute the time evolution of the distribution function and present
the numerical results in  Figures \ref{fig:t2-1} and \ref{fig:t2-2} where
computations were performed respectively with $n^3 = 16^3$, $n^3 = 24^3$ and $n^3=
32^3$  collocation points.

On the one hand, we present in Figure \ref{fig:t2-1}, the time evolution of the relative entropy
\eqref{h:rel} and the fourth order moment of the distribution function
\be
\label{eq:M4}
M_4(t) = \int_{\R^3} f(t,\bv) \,\dD \bv.
\ee
The relative entropy $\cH[f|\cM_{\rho, \bu,T}]$ and the fourth order moment $M_4$ are computed by
discretizing the expressions \eqref{h:rel} and \eqref{eq:M4} on the velocity grid by a
straightforward formula.

We compare these quantities  with a reference solution obtained with
$n^3=128^3$ collocation points. We observe that with only $n=24$ in
each direction, we get an accurate solution.

\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[width=8cm]{t2-f1.png} &
  \includegraphics[width=8cm]{t2-f2.png}
      \\
 (a) $\cH[f|\cM_{\rho, \bu,T}]$ & (b) $M_4[f]$
\end{tabular}
  \end{center}
\caption{{\bf Rosenbluth problem:} time evolution of the relative
  entropy \eqref{h:rel} and the fourth order moment \eqref{eq:M4}.}
\label{fig:t2-1}
\end{figure}

On the other hand, in Figure \ref{fig:t2-2} , we show the cross section of the distribution function at times $t = 0, 0.5, 1.5, 3, 5$, and
$50$. The results are in good agreement with those presented in
\cite{RMJ, PRT2} even if the time scale does not corresponds since we
take all physical constants equal to one.

\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[width=8.cm]{t2-f4.png} &
  \includegraphics[width=8.cm]{t2-f3.png}
      \\
 (a) $n^3=16^3$ & (b) $n^3=32^3$
\end{tabular}
  \end{center}
\caption{{\bf Rosenbluth problem:} time evolution the distribution
  function $f(t,0,0,v_z)$ for (a) $n^3=16^3$ and (b)  $n^3=32^3$ at
  time $t = 0, 0.5, 1.5, 3, 5$, and $50$.}
\label{fig:t2-2}
\end{figure}

\subsection{Trend to equilibrium for the sum of two Gaussians}
We now choose the initial condition as
$$
f_0(\bv)\,=\,\frac{1}{2(2\pi\sigma^2)^{3/2}}\,\left[
  \exp\left(-\frac{|\bv - 2 \sigma\ber_1|^2}{2\sigma^2}\right) \,+\ \exp\left(-\frac{|\bv - 2 \sigma \ber_1|^2}{2\sigma^2}\right)\right],
$$
with $\sigma = \pi/10$ and $\ber_1=(1,0,0)$. We choose $\Delta t=5.\,
10^{-3}$ when $n^3=32^3$ and the computational domain is $[-R,R]^3$
with $R=2.75$.

This test is used to
compute the evolution of the entropy and the pressure tensor defined by
\be
\label{eq:P}
\bP(t) \,=\, \int_{\R^3} (\bv-\bu)\otimes (\bv-\bu) \,f(t,\bv)\,\dD \bv\,,
\ee
where $\bu$ is  the mean velocity.

In this test, we compare the long time behavior of two numerical
solutions given by \eqref{eq:24} and by the steady state preserving
method \eqref{eq:24-bis} with the same numerical resolution.

In the Figures \ref{fig:t3-1} and \ref{fig:t3-2}, the dotted lines
represent the numerical solutions obtained with $32^3$ modes whereas  the
continuous line represent a reference solution using  $64^3$
modes. Both schemes give similar results when we observe the time
evolution of the relative entropy $\cH[f|\cM_{\rho, \bu,T}]$, which decreases to
zero, when $t$ becomes large.  However, passing to log
scale allows to observe a difference on the two solutions around the
equilibrium since the relative entropy corresponding to the solution
\eqref{eq:24} seems to saturate for $t \geq 1.5$.

Furthermore, in  Figure \ref{fig:t3-2}, we present the time evolution of the
pressure tensor $\bP$ and the fourth order moment. Again, the steady
state preserving method gives satisfying results when time become
large. Finally, in Figure \ref{fig:t3-3}, we propose several snapshots
of the projection of $f$ at different time,
$$
F(t,v_x,v_y) = \int_{\R} f(t,\bv)\,\dD v_z.
$$

\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[width=8cm]{t3-f1.png} &
  \includegraphics[width=8cm]{t3-f3.png}
      \\
 (a) $\cH[f|\cM_{\rho, \bu,T}]$ & (b) $\cH[f|\cM_{\rho, \bu,T}]$ in log scale
\end{tabular}
  \end{center}
\caption{{\bf Trend to equilibrium for the sum of two Gaussians: } time evolution of the relative
  entropy \eqref{h:rel}.}
\label{fig:t3-1}
\end{figure}

\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[width=8cm]{t3-f4.png} &
  \includegraphics[width=8cm]{t3-f2.png}
      \\
 (a) $\bP$ & (b) $M_4$
\end{tabular}
  \end{center}
\caption{{\bf Trend to equilibrium for the sum of two Gaussians: } time
  evolution of the Pressure tensor $\bP$ in \eqref{eq:P} and the fourth
  order moment $M_4$ given by \eqref{eq:M4}.}
\label{fig:t3-2}
\end{figure}

\begin{figure}[ht!]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[width=8.cm]{t3-f5.png} &
  \includegraphics[width=8.cm]{t3-f6.png}
      \\
  \includegraphics[width=8.cm]{t3-f7.png} &
  \includegraphics[width=8.cm]{t3-f8.png}
\end{tabular}
  \end{center}
\caption{{\bf Trend to equilibrium for the sum of two Gaussians: } time
  evolution the projection of the distribution
  function $f$ on the plane $(O, \,v_x, \,v_y)$ at different time.}
\label{fig:t3-3}
\end{figure}

\section{Conclusion.}
We have presented an efficient and accurate numerical method
to solve the space homogeneous Landau equation for plasma physics. The collision operator is solved in only $O(n^3\log_2 n)$ operations
using a spectral collocation method and discrete fast Fourier transforms to evaluate the derivatives and the non-local term. We take
advantage of the particular structure of Coulombian interactions to reduce the number of discrete convolutions.  This approach highly improves the efficiency of the method and seems
to be a good compromise between accuracy and computational cost. The scheme conserves mass, and approximate momentum and energy with
spectral accuracy provided that a sufficiently large support is the
velocity space is used.

This is a first step in the construction of an effective scheme for the
numerical solution of the Landau equation. We have seen that our method
is suitable for treating cases when the distribution function can be
effectively described with a reasonably low number of collocation
points, in particular, when the distribution
function is  smooth. For non-smooth solutions, this require more
investigations. In the near future we plan to extend this approach to
spatially nonhomogeneous situations following the previous works in
\cite{CF,DDFT} and also to find  a suitable time discretization to
avoid the restriction (parabolic CFL)  on the time step \cite{FJ,LM05}.

\section*{Acknowledgement}

The author is supported by the EUROfusion Consortium and has
received funding from the Euratom research and training programme
2014-2018 under grant agreement No 633053. The views and opinions
expressed herein do not necessarily reflect those of the European Commission.


TITLE: Does Lindeberg's condition imply $s_n \to \infty$?
QUESTION [1 upvotes]: Lindeberg's theorem  states that if we have a sequence of independent random variables $X_j: \Omega \to \mathbb{R}$ with zero mean, variance $\sigma_i^2$, and distribution $\alpha_i$, and we define $s_n^2 = \sigma_1^2 + \dots + \sigma_n^2$, then the distribution of $\frac{X_1+\dots +X_n}{s_n}$ converges weakly to the standard normal ${\bf if}$ for all $\varepsilon>0$,
$$
\lim_{n\to \infty} \frac{1}{s_n^2}\sum\limits_{i=1}^n \int\limits_{|x| > \varepsilon s_n} x^2 d\alpha_i(x) = 0. \qquad (\star)
$$
If $\phi_i$ is the characteristic function of some $X_i$, the proofs I see use some order of the Taylor expansion for $\log\phi_i\left(\frac{t}{s_n}\right)$ where $|t|<T$. I guess this tacitly assumes the $s_n$ diverges? However I could not verify if $(\star)$ implies this condition. Is this easy to see?

REPLY [1 votes]: Assume $s_n \to s$ with $s \in (0,\infty)$.
To start, suppose $\sigma_j>0$ for all $j$. Now look at $\varepsilon=\frac{1}{2} \frac{\sigma_1}{s}$. Then the first integral in the sum will remain bounded away from zero (since you will have $\varepsilon s_n \leq \frac{1}{2} \sigma_1$) and there will be no exploding factor in the denominator to send the whole thing to zero. So the Lindeberg condition will fail.
If $\sigma_1=0$, replace $\sigma_1$ in the above with whichever positive $\sigma_k$ you like.
If instead $s_n \to 0$ then $\sigma_j \equiv 0$, in which case the conclusion of the CLT doesn't hold (but this case is trivial, of course).


text:
\begin{document}

{\ }\qquad \hskip 4.3in
\vspace{.2in}

\begin{center} {\LARGE Noncommutative Differential Geometry and
Twisting of Quantum Groups}
\\ \baselineskip 13pt{\ }\\
{\ }\\ Shahn Majid \\ {\ }\\ School of Mathematical Sciences,
Queen Mary and Westfield College\\ University of London, Mile End
Rd, London E1 4NS\footnote{Reader and Royal Society University
Research Fellow}

\end{center}

\begin{quote}
\noindent{\bf Abstract}
We outline the recent classification of differential structures for
all main classes of quantum groups. We also outline the algebraic
notion of `quantum manifold' and `quantum Riemannian manifold' based on
quantum group principal bundles, a formulation that works over general
unital algebras.
 \end{quote}

\section{Introduction}

There have been many attempts in the last decades to arrive at a
theory of noncommutative geometry applicable to `coordinate'
algebras that are not necessarily commutative, notably that of A.
Connes coming out of abstract $C^*$-algebra theory in the light of
the Gelfand-Naimark and Serre-Swan theorems. One has tools such as
cyclic cohomology and examples such as the noncommutative torus
and other foliation $C^*$-algebras. Another `bottom up' approach,
which we outline, is based on the idea that the theory should be
guided by the inclusion of the large vein of `naturally occuring'
examples, the coordinate algebras of the quantum groups $U_q(g)$
in particular, and Hopf algebras in general, whose validity for
several branches of mathematics has already been established. This
is similar to the key role that Lie groups played in the
development of modern differential geometry. Much progress has
been made in recent years and there is by now (at least at the
algebraic level) a more or less clear formulation of `quantum
manifold' suggested by this approach. After being validated on the
q-deformation examples such as quantum groups, quantum homogeneous
spaces etc, one can eventually apply the theory quite broadly to a
wide range of unital algebras. The approach will be algebraic,
although not incompatible with $C^*$ completions at a later stage.
In particular, as a bonus, one can apply the theory to
finite-dimensional algebra, i.e. to discrete classical and quantum
systems.

An outline of the paper is the following. We start with the lowest
level structure which (in our approach) is the choice of
differential structure. This is the topic of Section~2 where we
outline the recently achieved more or less complete classification
results. In Section~3 we develop the notion of `quantum
manifold'\cite{Ma:rie} based on noncommutative frame bundles with
quantum group fibre. Usual ideas of `sheaf theory' and `local
trivialisations' do not work in this setting, but from
\cite{BrzMa:gau} one has global algebraic replacements. There is also
an appropriate notion of automorphism or diffeomorphism quantum
groups\cite{Ma:dif}.

\section{Quantum differential forms}

Let $M$ be a unital algebra, which we consider as playing the role of
`co-ordinates' in algebraic geometry, except that we do not require
the algebra to be commutative. The appropriate notion of cotangent
space or differential 1-forms in this case is

1. $\Omega^1$ an $M$-bimodule

2. $\extd:M\to \Omega^1$ a linear map obeying the Leibniz rule
$\extd(ab)=a\extd b+(\extd a)b$ for all $a,b\in M$.

3. The map $M\tens M\to \Omega^1$, $a\tens b\mapsto a\extd b$ is
surjective.

Differential structures are not
unique even classically, and even more non-unique in the quantum
case. There is, however, one universal example of which others are
quotients. This is
\eqn{univ}{ \Omega^1_{\rm univ}=\ker\cdot\subset M\tens M,\quad \extd
a=a\tens 1-1\tens a.}
This is common to more or less all approaches to noncommutative geometry.

The main feature here is that, in usual algebraic geometry, the
multiplication of forms $\Omega^1$ by `functions' $M$ is the same
from the left or from the right. However, if $a\extd b=(\extd b)a$
then by axiom 2. we have $\extd(ab-ba)=0$, i.e. we cannot naturally
suppose this when $M$ is noncommutative.  We say that a
differential calculus is noncommutative or `quantum'  if the left
and right multiplication of forms by functions do not coincide.

When $M$ has a Hopf algebra structure with coproduct $\Delta:M\to
M\tens M$ and counit $\eps:M\to k$ ($k$ the ground field), we say
that $\Omega^1$ is {\em bicovariant} if\cite{Wor:dif}

4. $\Omega^1$ is a bicomodule with coactions $\Delta_L:\Omega^1\to
M\tens\Omega^1,\Delta_R:\Omega^1\to \Omega^1\tens M$ bimodule maps (with
the tensor product bimodule structure on the target spaces, where
$M$ is a bimodule by left and right multiplication).

5. $\extd$ is a bicomodule map with the left and right regular
coactions on $M$ provided by $\Delta$.

A morphism of calculi means a bimodule and bicomodule map forming a
commuting triangle with the respective $\extd$ maps. One
says\cite{Ma:cla} that a calculus is {\em coirreducible} if it has no
proper quotients. Whereas the translation-invariant calculus is unique
classically, in the quantum group case we have at least complete
classification results in terms of representation theory\cite{Ma:cla}.
The dimension of a calculus is that of its space of (say)
left-invariant 1-forms, which can be viewed as generating the rest of
the calculus as a right $M$-module. Similarly with left and right
interchanged.

We note also that in the bicovariant case there is a natural
extension\cite{Wor:dif} from $\Omega^1$ to $\Omega^n$ with $\extd^2=0$.
This is defined as the tensor algebra over $M$ generated by $\Omega^1$
modulo relations defined by a braiding which acts by a simple transposition
on left-invariant and right-invariant forms. Other extensions are also
possible and in general the differential structure
can be specified order by order. Given the extension, one has a quantum DeRahm
cohomology defined in the usual way as closed forms modulo exact ones. Apart
from cohomology one can also start to do `$U(1)$' gauge theory with trivial
bundles, where a gauge field is just a differential form $\alpha\in \Omega^1$
and its curvature is $F=\extd\alpha+\alpha\wedge\alpha$, etc. A gauge transform
is \eqn{trivialgauge}{ \alpha^\gamma=\gamma^{-1}\alpha\gamma
+\gamma^{-1}\extd \gamma,\quad F^\gamma=\gamma^{-1} F\gamma} for any
invertible `function' $\gamma\in M$, and so on. One can define then the
space of flat connections as those with $F=0$ modulo gauge
transformation.  This gives two examples of `geometric' invariants
which work therefore for general algebras equipped with differential
structure.

\subsection{$M=k[x]$}

For polynomials in one variable the
coirreducible calculi have the
form\cite{Ma:fie}
\[ \Omega^1=k_\lambda[x],\quad  \extd
f(x)=\frac{f(x+\lambda)-f(x)}{\lambda},\quad f(x)\cdot
g(\lambda,x)=f(x+\lambda)g(\lambda,x),\quad g(\lambda,x)\cdot
f(x)=g(\lambda,x)f(x)\]
for functions $f$ and one-forms $g$.  Here $k_\lambda$ is a field
extension of the form $k[\lambda]$ modulo $m(\lambda)=0$ and $m$ is an
irreducible monic polynomial.  The dimension of the calculus is the
order of the field extension or the degree of $m$.

For example, the calculi on $\C[x]$ are classified by $\lambda_0\in
\C$ (here $m(\lambda)=\lambda-\lambda_0$) and one has
\[ \Omega^1=\extd x\C[x],\quad \extd f=\extd
x\frac{f(x+\lambda_0)-f(x)}{\lambda_0},\quad x\extd x=(\extd
x)x+\lambda_0.\] We see that the Newtonian case $\lambda_0=0$ is
only one special point in the moduli space of quantum differential
calculi. But if Newton had not supposed that differentials and
forms commute he would have had no need to take this limit. What
one finds with noncommutative geometry is that there is no need to
take this limit at all.  It is also
interesting that the most important field extension in physics,
$\R\subset\C$, can be viewed noncommutative-geometrically with
complex functions $\C[x]$ the quantum 1-forms on the algebra of
real functions $\R[x]$. There is nontrivial quantum DeRahm cohomology
in this case.

\subsection{$M=\C[G]$}

For the coordinate algebra of a finite group $G$ (for convenience
we work over $\C$)  the coirreducible calculi correspond to nontrivial
conjugacy classes $\CC\subset G$ and have the form
\[ \Omega^1=\CC\cdot\C[G],\quad \extd f
=\sum_{g\in\CC}g\cdot (L_g(f)-f),\quad f\cdot g=g\cdot L_g(f)\]
where $L_g(f)=f(g\cdot)$ is the translate of $f\in \C[G]$. The dimension
of the calculus is the order of the conjugacy class.

For the coordinate algebra $\C[G]$ of a Lie group with Lie algebra
$\cg$ the coirreducible calculi  correspond to maximal ideals in
$\ker\eps$ stable under the adjoint coaction. Or in a natural
reformulation\cite{Ma:cla} in terms of quantum tangent spaces the
correspondence is with irreducible $Ad$-invariant subspaces of the
enveloping algebra $\ker\eps\subset U(\cg)$ which are stable under the
coaction $\Delta_L=\Delta-\id\tens 1$ of $U(\cg)$. For example $\cg$
itself defines the standard translation-invariant calculus and this is
coirreducible when $\cg$ is semisimple.

\subsection{$M=\C G$}

For the group algebra of a nonAbelian finite group $G$, we definitely
need the machinery of noncommutative geometry since $M$ itself is
noncommutative. We regard these group algebras `up side down' as if
coordinates, i.e. we describe the geometry of the noncommutative space
$\hat G$ in some sense. The above definitions make sense and
differential structures abound.  The coirreducible calculi correspond
to pairs $(V,\rho,\lambda)$ where $(V,\rho)$ is a nontrivial
irreducible representation and $\lambda\in V/\C$\cite{Ma:cla}. They
have the form \[ \Omega^1=V\cdot \C G,\quad \extd  g
=((\rho(g)-1)\lambda)\cdot g,\quad g\cdot v=(\rho(g)v)\cdot g\] where
$g\in G$ is regarded as a `function'. The dimension of the calculus is
that of $V$. The minimum assumption for merely a differential calculus
is that $\lambda$ should be cyclic.

For $M=U(\cg)$ (the Kirillov-Kostant quantisation of $\cg^*$) one has a
similar construction for any irreducible representation $V$ of the Lie
algebra $\cg$ and choice of ray $\lambda$ in it. Then \[\Omega^1=V\cdot
U(\cg),\quad \extd \xi=\rho(\xi)\lambda,\quad \xi\cdot
v=\rho(\xi)v+v\cdot\xi\] where $\xi\in\cg$ is regarded as a `function'.
The dimension is again that of $V$.

For example, let $\cg=b_+$ be the 2-dimensional Lie algebra with
$[x,t]=x$. Let $V$ be the 2-dimensional representation with matrix and
ray vector
\[ \rho(t)=\begin{pmatrix}0&0\\ 0&1\end{pmatrix},\quad
\rho(x)=\begin{pmatrix}0&1\\ 0&0\end{pmatrix},\quad
\lambda=\begin{pmatrix}0\\ 1\end{pmatrix}.\]
Then $\extd t=\lambda$ and $\extd x$ are the usual basis of $V$ and obey
\[ [t,\extd x]=[x,\extd x]=0,\quad [t,\extd t]=\extd t,\quad
[x,\extd t]=\extd x.\] Replacing $x$ trivially by a vector $x_i$,
$i=1,2,3$ one obtains similarly a natural candidate for noncommutative
Minkowski space along with its differential structure. It has measurable
astronomical predictions\cite{AmeMa:wav}.

This covers the classical objects or their duals viewed `up side down'
as noncommutative spaces. For a finite-group bicrossproduct
$\C[M]\bicross \C G$ the classification is a mixture of the two cases
above and is given in \cite{BegMa:dif}. The Lie version remains to be
worked out in detail. The important example of the Planck scale quantum
group $\C[x]\bicross\C[p]$, however, is a twisting by a cocycle of its
classical limit $\C[\R\rcross\R]$ and is therefore covered by a later
subsection.

\subsection{Proofs} The above cases are all sufficiently elementary
that they can be easily worked out using the following simple
observations known essentially (in some form or other) since
\cite{Wor:dif}. We suppose for convenience that $H$ has invertible
antipode.

1. $\ker\eps\subset M$ is an object in the braided category of left
crossed $M$-modules (i.e. modules over the quantum double $D(M)$ in the
finite-dimensional case)  by multiplication and the left adjoint
coaction.

2. The isomorphism $M\tens M\isom M\tens M$ given by $a\tens b\mapsto
(\Delta a)b$ restricts to an isomorphism $\Omega^1_{\rm Univ}\isom
\ker\eps \tens M$ of bimodules and of bicomodules, where the right hand
side is a right (co)module by the (co)product of $M$ and a left
(co)module by the tensor product of the (co)action on $\ker\eps$ and
the (co)product of $M$.

This implies that every other bicovariant $\Omega^1$ is of the form
$\Omega^1\isom \Omega_0\tens M$ where $\Omega_0$ is a quotient object
of $\ker\eps$ in the category of crossed $M$-modules. I.e. the calculi
correspond to ideals in $\ker\eps$ stable under the adjoint coaction.
Given $\Omega^1$ the space $\Omega_0$ is given by the right-invariant
differentials. In categorical terms the braided category of bicovariant
$M$-bimodules as featuring above (i.e. bimodules which are also
bicomodules with structure maps being bimodule maps) can be identified
with that of crossed $M$-modules, under which the Hopf module for the
universal calculus corresponds to $\ker\eps$.

When this is combined with the notion of coirreducibility and with the
Peter-Weyl decomposition of an appropriate type for $\ker\eps$, one
obtains the classification results above. These latter steps have been
introduced by the author\cite{Ma:cla} (before that one found only
sporadic examples of calculi on particular quantum groups, usually
close to the unique classical calculus.)

We also note that for any finite-dimensional bicovariant calculus the
map $\extd:M\to \Omega_0\tens M$ can be viewed as a `partial
derivative' $\del_x:M\to M$ for each $x\in \Omega_0^*$. The space
$\Omega_0^*$ is called the invariant `quantum tangent space' and is
often more important than the 1-forms in applications. These $\del_x$
are not derivations but together form a braided derivation in the
braided category of $M$-crossed modules (there is a braiding as $x\in
\Omega_0^*$ passes $a\in M$) \cite{Ma:cla}.

\subsection{Cotriangular quantum groups and twisting of calculi}

We recall\cite{Ma:book} that if $M$ is a quantum group and $\chi:M\tens
M\to k$ a cocycle in the sense
\[ \chi(b\o\tens c\o)\chi(a\tens b\t c\t)=\chi(a\o\tens b\o)
\chi(a\t b\t\tens c),\quad \chi(1\tens a)=\eps(a),\quad\forall a,b,c\in M\]
then there is a `twisted' quantum
group $M^\chi$ with product \[ a\cdot_\chi b=\chi(a\o\tens b\o)a\t b\t
\chi^{-1}(a\th\tens b\th)\] and unchanged unit, counit and coproduct.
Here $\chi^{-1}$ is the inverse in $(M\tens M)^*$, which we assume, and
$\Delta a=a\o\tens a\t$, etc., is a notation.

\begin{theorem}\cite{MaOec:twi} The bicovariant differentials
$\Omega^1(M^\chi)$ are in 1-1 correspondence with those of $M$.
\end{theorem}
In fact the entire exterior algbera in the bicovariant
case is known to be a super-Hopf algebra (Brzezinski's theorem) and
that of $M^\chi$ is the twist of that of $M$ when $\chi$ is trivially
extended to a cocycle on the latter. The more direct proof involves the
following:

\begin{theorem}\cite{MaOec:twi} There is an equivalence $\CF$ of braided
monoidal categories from left $M$-crossed modules to left
$M^\chi$-crossed modules given by the functor
\[ \CF(V,\la,\Delta_L)=(V,\la^\chi,\Delta_L),\quad a\la^\chi v
=\chi(a\o\tens v\bo)(a\t\la v\bt)\bt\chi^{-1}((a\t\la v\bt)
\bo\tens a\th),\quad \forall a\in M,\ v\in V,\]
where $\la$ denotes the action and $\Delta_Lv=v\bo\tens v\bt$ is a
notation. There is an associated natural transformation
\[ c_{V,W}: \CF(V)\tens \CF(W)\isom \CF(V\tens W),\quad c_{V,W}(v\tens w)
=\chi(v\bo\tens w\bo)v\bt\tens w\bt.\]
\end{theorem}

As a corollary we deduce by Tannaka-Krein reconstruction arguments:

\begin{corol}\cite{MaOec:twi} When $M$ is finite-dimensional the dual
of the Drinfeld double $D(M^\chi)$ is isomorphic to a twist  of the
dual of the Drinfeld double $D(M)$ by $\chi^{-1}$ viewed on $D(M)\tens
D(M)$. \end{corol}

Here we use the theorem\cite{Ma:tan} that an equivalence of comodule
categories respecting the forgetful functor corresponds to a twist of
the underlying quantum groups. The corollary itself can then be
verified directly at an algebraic level once the required (nontrivial)
isomorphism has been found in this way. Of course one can state it also
in terms of Drinfeld's coproduct twists.

Starting with a classical (commutative) Hopf algebra such a twist
yields a cotriangular one and (from recent work of Etingof and
Gelaki\cite{EtiGel}) every finite-dimensional cotriangular Hopf algebra
in the (co)semisimple case over $k$ algebraically closed is of this
form. Hence the differential calculus in this case reduces by the above
theorem to the classification in the classical cases considered in
previous sections. There are many other instances where an important
quantum group is a twisting of another -- the theorem provides its
differential calculus from that of the other.

\subsection{Factorisable quantum groups $M=\C_q[G]$}

Finally, we come to the standard quantum groups $\C_q[G]$ dual to the
Drinfeld-Jimbo $U_q(\cg)$.  Here\cite{Ma:cla} the coirreducible calculi
are essentially provided by nontrivial finite-dimensional irreducible
right comodules $V$ of the quantum group (i.e. essentially by the
irreducible representations of the Lie algebra) and have the form
\eqn{diffqG}{ \Omega^1=\End(V)\cdot\C_q[G],\quad \extd a
=\rho_+(a\o)\circ \rho_-(Sa\t)\cdot a\th-\id\cdot a,\quad
a\cdot\phi=\rho_+(a\o)\circ\phi\circ \rho_-(S a\t)\cdot a\th }
for all $\phi\in\End(V)$, where
\[ \rho_+(a)v=v\bo\CR(a\tens v\bt),\quad \rho_-(a)v
=v\bo\CR^{-1}(v\bt\tens a)\]
and $\CR:\C_q[G]\tens \C_q[G]\to \C$ is the dual-quasitriangular
structure\cite{Ma:book}. $S$ denotes the antipode. The construction
works for any dual-quasitriangular Hopf algebra with factorisable $\CR$
(the minimum one needs for a differential calculus is that
$\CQ_\rho(a)=\rho_+(a\o)\rho_-(Sa\t)$ is surjective) and gives a
classification of calculi if $M$ has in addition the Peter-Weyl
property that $M=\oplus_V V\tens V^*$ as left and right $M$-comodules.
This is the result in \cite[Thm. 4.3]{Ma:cla} cast in a purely comodule
form. Or in the original formulation in terms of quantum tangent spaces
the correspondence is given in more familiar terms with representations
$\rho$ of the quantum enveloping algebra and
\[ \rho_+(a)=(a\tens\rho)(\CR),\quad \rho_-(a)=(\rho\tens a)(\CR^{-1}),\]
where $\CR$ is the quasitriangular structure or universal R-matrix of
$U_q(\cg)$. For finite-dimensional representations only a finite number
of terms in its powerseries contribute here, i.e. there are no
convergence problems.

The factorisability etc. hold formally for $\C_q[G]$ so that although
one has one natural calculus for each irreducible representation there
are also some `shadows' or technical variants allowed according to the
precise formulation of the relevant quantum groups and their duality
(this is more a deficit in the technical definitions than anything
else). The latter aspect has been subsequently clarified in
\cite{BauSch}\cite{HecSch} following our basic result in
\cite{Ma:cla}.

For the sake of a sketch of the proof of the basic result assume that
$M$ is strictly factorisable dual-quasitriangular and has Peter-Weyl
decomposition in terms of irreducible representations $(V,\rho)$ of a
suitable dual Hopf algebra $U$.  Classifying the quotient $M$-crossed
modules of $\ker\eps$ is equivalent essentially to classifying the
subobjects of $\ker\eps\subset U$ as $U$-crossed modules (the quantum
tangent spaces). When $U$ is strictly factorisable its quantum double
$D(U)$ is isomorphic to $U\codcross U$ which, as an algebra, is a
tensor product (the coproduct is twisted). Hence $U$-crossed modules
are equivalent to two $U$-modules. Next, under the isomorphism of
linear spaces $U\isom M$ assumed under strict factorisability, this is
the same as classifying subobjects of $\ker\eps\subset M$. The
$U$-crossed module structure on this final $\ker\eps\subset M$ under
this chain of reasoning is simply evaluation against $M$ coacting
independently from the left and the right (viewed from the left via the
antipode). This is just the action with respect to which the assumed
Peter-Weyl decomposition $M=\oplus_V \End(V)$ is a decomposition into
irreducibles as $V$ runs over the irreducible representations of $U$.
One may make a similar proof working only with $M$-crossed modules
and $M$-comodules throughout and the corresponding comodule Peter-Weyl
decomposition.

\begin{propos}
\cite{Ma:cla} The quantum tangent spaces $\Omega_0^*=V^*\tens V$ for
the above differential calculi on $\C_q[G]$ are braided-Lie algebras
in the sense of \cite{Ma:lie}. The action of basis element $f^i\tens
e_i$ is
\[ \del_{x^i{}_j} (a)=\< x^i{}_j,a\o\>a\t,\quad x^i{}_j
=\CQ(\rho^i{}_j)-\delta^i{}_j\]
where $\CQ:\C_q[G]\to U_q(\cg)$ is defined by $\CR_{21}\CR$ and
$\rho^i{}_j\in \C_q[G]$  are the matrix elements in the representation
$V$ with basis $\{e_i\}$ and dual basis $\{f^i\}$.
\end{propos}

Recall that the dual of any quantum group acts on the quantum group by
the `coregular representation' in the manner shown, in our case by the
$x^i{}_j$. These and $1$ together form a braided-Lie algebra. This is
described by a system of axioms in any braided category including a
pentagonal `braided-Jacobi' identity. Moreover, such objects have
braided enveloping algebras which, for usual Lie algebras $\cg$,
returns a homogenized version of $U(\cg)$. In our case it returns a
quadratic and braided version of $U_q(\cg)$, i.e. this solved (some
years ago\cite{Ma:lie}) the Lie problem for such quantum groups. The
above gives its geometric interpretation.

For the example of $\C_q[SU_2]$ there is basically one bicovariant
calculus for each spin $j$ with dimension $(2j+1)^2$. The lowest
corresponds to the 4-dimensional braided-Lie algebra $gl_{q,2}$ spanned
by
\[ x=\begin{pmatrix}q^H-1& q^{-\h}(q-q^{-1})q^{H\over 2}X_-\\
q^{-\h}(q-q^{-1})X_+q^{H\over 2}&
q^{-H}-1+q^{-1}(q-q^{-1})^2X_+X_-\end{pmatrix} \]
in the usual notations for $U_q(su_2)$. This braided-Lie algebra is
irreducible for generic $q$ but as $q\to 1$ it degenerates into
$su_2\oplus u(1)$. The partial derivatives degenerate into the usual
invariant vector fields on $SU_2$ and an additional 2nd order operator
which turns out to be the Casimir or Laplacian.

\subsection{Discrete manifolds}

To close with one non-quantum group example, consider
any actual manifold with a finite good cover $\{U_i\}_{i\in I}$.
Instead of building geometric invariants on a manifold and
studying them modulo diffeomorphisms we can use the methods above
to first pass to the skeleton of the manifold defined by its open
set structure and do differential geometry directly on this
indexing set $I$. Thus we take $M=\C[I]$ which just means
collections $\{f_i\in \C\}$. The universal $\Omega^1$ is just
matrices $\{f_{ij}\}$ vanishing on the diagonal. We use the
intersection data for the open sets to set some of these to zero.
Similarly for higher forms. Thus\cite{BrzMa:dif}
\[\Omega^1=\{f_{ij}|\ U_i\cap U_j\ne \emptyset\},\quad
\Omega^2=\{f_{ijk}|\ U_i\cap U_j\cap U_k\ne\emptyset\}\]
\[(\extd f)_{ij}=f_i-f_j,\quad (\extd f)_{ijk}
=f_{ij}-f_{ik}+f_{jk}\]
and so on. Then one has that the quantum cohomology is just the
additive Cech cohomology of the original manifold. Similarly,
 one has that the zero curvature gauge
fields modulo gauge transformations recovers again the first Cech
cohomology, but now in a multiplicative form.

\section{Bundles and connections}

The next layer of differential geometry is bundles, connections, etc.
Usually in physics one needs only the local picture with trivial
bundles in each open set -- but for a general noncommutative algebra
$M$ there may be no reasonable `open sets' and one has therefore to
develop the global picture from the start. This is needed for example
to describe the frame bundle of a topologically nontrivial `manifold'.
It also turns out to be rather easier to do the gauge theory beyond the
`$U(1)$' case (i.e. with a nontrivial quantum structure group and
nonuniversal calculus on it) if one takes the global point of view,
even if the bundle itself is trivial. We take a Hopf algebra $H$ in the
role of `functions' on the structure group of the bundle. To keep
things simple we concentrate on the universal differential calculus but
it is important that the general case is also covered by making
suitable quotients.  Recall that a classical bundle has a free action
of a group on the total space $P$ and a local triviality property. In
our algebraic terms we need \cite{BrzMa:gau}:

1. An algebra $P$ and a coaction $\Delta_R:P\to P\tens H$ of the
quantum group $H$ such that the fixed subalgebra is $M$,
\eqn{fixed}{ M=P^H=\{p\in P|\ \Delta_R p=p\tens 1\}.}
We assume that $P$ is flat as an $M$-module.

2.  The sequence
\eqn{exactness}{ 0\to P(\Omega^1M)P\to\Omega^1P{\buildrel{\rm ver}
\over\longrightarrow} P\tens \ker\eps\to 0}
is exact, where ${\rm ver}=(\cdot\tens\id)\Delta_R$.

The map ${\rm ver}$ plays the role of generator of the vertical vector
fields corresponding classically to the action of the group (for each
element of $H^*$ it maps $\Omega^1P\to P$ like a vector field).
Exactness on the left says that the one-forms $P(\Omega^1M)P$ lifted
from the base are exactly the ones annihilated by the vertical vector
fields.  In the universal calculus case this can be formulated as a
Hopf-Galois extension, a condition arising in other contexts in Hopf
algebra theory also. The differential geometric picture is more
powerful and includes general calculi when we use the right-handed
version of $\Omega_0$ in place of $\ker\eps$.

One can then define a connection as an equivariant splitting
\eqn{connection}{ \Omega^1P=P(\Omega^1 M)P\oplus {\rm complement}}
i.e. an equivariant projection $\Pi$ on $\Omega^1P$. One can
show\cite{BrzMa:gau} the required analogue of the usual theory, i.e.
that such a projection corresponds to a connection form such that
\eqn{conform}{\omega:\ker\eps\to\Omega^1P,\quad {\rm ver}\circ
\omega=1\tens\id}
where $\omega$ intertwines with the adjoint coaction of $H$ on
itself. Finally,  if $V$ is a
vector space on which $H$ coacts then we define the associated
`bundles' $E^*=(P\tens V)^H$ and $E=\hom^H(V,P)$, the space of
intertwiners. The two bundles should be viewed geometrically as
`sections' in classical geometry of bundles associated to $V$ and
$V^*$. Given a suitable (so-called strong) connection one has a covariant
derivative\cite{BrzMa:gau}
\eqn{covderiv}{ D_\omega:E\to \Omega^1 M\tens_M E,\quad D_\omega
=(\id-\Pi)\circ \extd.}

All of this can be checked out for the example of
the $q$-monopole bundle over the
$q$-sphere\cite{BrzMa:gau}. Recall that classically the inclusion
$U(1)\subset SU_2$ in the diagonal has coset space $S^2$ and
defines the $U(1)$ bundle over the sphere on which the monopole
lives. In our case the coordinate algebra of $U(1)$
is the polynomials $\C[g,g^{-1}]$ and the classical inclusion
becomes the projection
\[\pi:\C_q[SU_2]\to \C[g,g^{-1}],\quad
\pi\left(\begin{matrix}
a&b\\ c&d
\end{matrix}\right)=\left(\begin{matrix}
g&0\\ 0&g^{-1}
\end{matrix}\right)\]
Its induced coaction $\Delta_R=(\id\tens\pi)\Delta$ is by the
degree defined as the number of $a,c$ minus the number of $b,d$ in
an expression. The quantum sphere $\C_q[S^2]$ is the fixed
subalgebra i.e. the degree zero part. Explicitly, it is generated
by $b_3=ad$, $b_+=cd$, $b_-=ab$ with $q$-commutativity relations
\[ b_\pm b_3=q^{\pm 2}b_3b_\pm+(1-q^{\pm 2})b_\pm,
\quad q^{2}b_-b_+=q^{-2}b_+b_-+(q-q^{-1})(b_3-1)\]
and the sphere equation $b_3^2=b_3+qb_-b_+$. When $q\to 1$ we can
write $b_\pm=\pm(x\pm\imath y)$, $b_3=z+\h$ and the sphere
equation becomes $x^2+y^2+z^2=\frac{1}{4}$ while the others become
that $x,y,z$ commute. One may verify that we have a quantum bundle
in the sense above and that there is a connection
$\omega(g-1)=d\extd a-qb\extd c$ which, as $q\to 1$, becomes the
usual Dirac monopole constructed algebraically. If we take $V=k$
with coaction $1\mapsto 1\tens g^n$, the sections of the
associated vector bundles $E_n$ for each charge $n$ are just the
degree $n$ parts of $\C_q[SU_2]$. The associated covariant
derivative acts on these.

This example demonstrates compatibility with the more traditional
$C^*$-algebra approach of A. Connes\cite{Con:geo} and others.
Traditionally a vector bundle over any algebra is defined as a finitely
generated projective module. However, there was no notion of quantum
principal bundle before quantum groups.

\begin{propos}\cite{HajMa:pro} The
associated bundles $E_n$ for the $q$-monopole bundle
are finitely generated projective modules, i.e. there exist
\[ e_n\in M_{|n|+1}(\C_q[S^2]),\quad e_n^2=e_n,\quad E_n
=e_n\C_q[S^2]^{|n|+1}.\]
The covariant derivative for the monopole has the form $e_n\extd e_n$.
The classes $[e_n]$ are elements of the noncommutative $K$-theory
$K_0(\C_q[S^2])$ and have nontrivial duality pairing with cyclic
cohomology, hence the $q$-monopole bundle is nontrivial.
\end{propos}

The potential applications of quantum group gauge theory hardly
need to be elaborated.  For example, for a classical manifold
\eqn{pi1}{ \left\{ {{{\rm Flat\ connections\ on}\ G-{\rm bundle}}
\atop{\rm modulo\ gauge}}\right\}\isom \hom(\pi_1,G)/G}
using the holonomy. One can view this as a functor from groups to sets
and the homotopy group $\pi_1$ as more or less the representing object
in the category of groups. The same idea with quantum group gauge
theory essentially defines a homotopy quantum group $\pi_1(M)$ for any
algebra $M$ as more or less the representing object of the functor that
assigns to a quantum group $H$ the set of zero-curvature gauge fields
with this quantum structure group.  This goes somewhat beyond vector
bundles and $K$-theory alone.  Although in principle defined, this idea
has yet to be developed in a computable form.

Finally we mention that one needs to make a slight generalisation of
the above to include other noncommutative examples of interest.  In
fact (and a little unexpectedly) the general theory above can be
developed with only a coalgebra rather than a Hopf algebra $H$.  Or
dually it means only an algebra $A$ in place of the enveloping algebra
of a Lie algebra. This was achieved more recently, in
\cite{BrzMa:coa}\cite{BrzMa:geo}, and allows us to include the full
2-parameter quantum spheres as well as (in principle)  all known
$q$-deformed symmetric spaces.  This setting of gauge theory based on
inclusions of algebras could perhaps be viewed as an algebraic analogue of the
notion of `paragroup' in the theory of operator algebras. Also, in a
different direction, one may do the quantum group gauge theory in any
braided monoidal category at the level of braids and
tangles\cite{Ma:diag} so that one has braided group gauge theory and in
principle gauge theory for quasiassociative algebras such
as\cite{AlbMa:qua} the octonions.

\section{Quantum soldering forms and metrics}

We are finally ready to take the plunge and offer at least a first
definition of a `quantum manifold'.   The approach we take is basically
that of the existence of a bein or, in global terms, a soldering form.
The first step is to define a generalised frame bundle or {\em frame
resolution} of our algebra $M$ as\cite{Ma:rie}

1. A quantum principal bundle
$(P,H,\Delta_R)$ over $M$.

2. A comodule $V$ and an equivariant
`soldering form' $\theta:V\to P\Omega^1M\subset\Omega^1P$ such
that the induced map
\eqn{frame}{ E^*\to \Omega^1M,\quad p\tens v\mapsto p\theta(v)}
is an isomorphism.

What this does is to express the cotangent
bundle as associated to a principal one. Other tensors are then
similarly associated, for example vector fields are
$E\isom\Omega^{-1}M$. Of course, all of this has to be done with
suitable choices of differential calculi on $M,P,H$ whereas we
have been focusing for simplicity on the universal calculi. There
are some technicalities here but more or less the same definitions
work in general. The working definition\cite{Ma:rie} of a {\em
quantum manifold} is simply this data
$(M,\Omega^1,P,H,\Delta_R,V,\theta)$. The definition works in that
one has analogues of many usual results. For example, a connection
$\omega$ on the frame bundle induces a covariant derivative
$D_\omega$ on the associated bundle $E^*$ which maps over under
the soldering isomorphism to a covariant derivative
\eqn{nabla}{ \nabla:\Omega^1M\to \Omega^1M\tens_M\Omega^1M.}
Its torsion is defined as corresponding similarly to
$\bar D_\omega\theta$, where we use a suitable (right-handed)
version of the covariant derivative.

Defining a Riemannian structure can be done in a `self-dual' manner as
follows. Given a framing, a `generalised metric' isomorphism
$\Omega^{-1}M\isom\Omega^1M$ between vector fields is equivalent
to\cite{Ma:rie}

3. Another framing $\theta^*:V^*\to(\Omega^1M)P$, which we
call the {\em coframing}, this time with $V^*$.

The associated quantum metric is
\eqn{quametric}{ g= \theta^*(f^a)\theta(e_a)\in\Omega^1M
\tens_M\Omega^1M}
where $\{e_a\}$ is a basis of $V$ and $\{f^a\}$ is a dual basis.

Now, this self-dual formulation of `metric' as framing and coframing is
symmetric between the two. One could regard the coframing as the
framing and vice versa. From our original point of view its torsion
tensor corresponding to $D_\omega\theta^*$ is some other tensor, which
we call the {\em cotorsion tensor}. A natural proposal for a
generalised Levi-Civita connection on a quantum Riemannian manifold is
therefore\cite{Ma:rie}

4. A connection $\omega$ such that the torsion and cotorsion
tensors both vanish.

There is a corresponding covariant derivative $\nabla$. The Riemannian
curvature of course corresponds to the curvature of $\omega$, which is
$\extd\omega+\omega\wedge\omega$, via the soldering form. I would not
say that the Ricci tensor and Einstein tensor are understood abstractly
enough in this formalism but of course one can just write down the
relevant contractions and proceed blindly.

\begin{theorem}\cite{Ma:rie}
Every quantum group $M$ has a framing by $H=M$, $P=M\tens M$,
$V=\ker\eps$ and $\theta$ induced from the quantum group Maurer-Cartan
form $e(v)=Sv\o\tens v\t$. Likewise for all $M$ equipped with a
bicovariant differential calculus, with $V=\Omega_0$.
\end{theorem}

In this construction one builds the framing from a $V$-bein $e$
inducing the isomorphism $\Omega^1(M)=M\tens\Omega_0$ as in Section~2
(in a right-handed setting). Moreover, for quantum groups such as
$\C_q[SU_2]$ there is an Ad-invariant non-degenerate braided Killing
form\cite{Ma:lie} on the underlying braided-Lie algebra,
 which provides a coframing from a framing -- so that quantum groups
such as $\C_q[SU_2]$ with the corresponding bicovariant differential
calculi are quantum Riemannian manifolds in the required sense. The
existence of a generalised Levi-Civita connection in such cases remains
open and may require one to go beyond strong connections.

At least with the universal calculus every quantum homogeneous space is
a quantum manifold too. That includes quantum spheres, quantum planes
etc. In fact, there is a notion of comeasuring or quantum automorphism
bialgebra\cite{Ma:dif} for practically any algebra $M$ and when this
has an antipode (which typically requires some form of completion) one
can write $M$ as a quantum homogeneous space. So almost any algebra $M$
is more or less a quantum manifold for some principal bundle (at least
rather formally).  This is analogous to the idea that any classical
manifold is, rather formally, a homogeneous space of diffeomorphisms
modulo diffeomorphisms fixing a base point.

Finally, to get the physical meaning of the cotorsion tensor and
other novel ideas coming out of this noncommutative Riemannian geometry,
let us consider the semiclassical limit. What we find is that
noncommutative geometry forces us to slightly generalise
conventional Riemannian geometry itself \cite{Ma:rie}:

1. We should allow any group $G$ in the `frame bundle', hence
the more general concept of a `frame resolution'
$(P,G,V,\theta_\mu^a)$ or {generalised manifold}.

2. The {generalised metric} $g_{\mu\nu}
=\theta^*_\mu{}^a\theta_{\nu a}$
corresponding to a coframing $\theta^*_{\mu}{}^a$ is nondegenerate
but need not be symmetric.

3. The {generalised Levi-Civita} connection defined as having
vanishing torsion and vanishing cotorsion respects the metric only
in a skew sense
\eqn{genlev}{ \nabla_\mu g_{\nu\rho}-\nabla_\nu g_{\mu\rho}=0}
and need not be uniquely determined.

This generalisation of Riemannian geometry includes special cases
of symplectic geometry, where the generalised metric is totally
antisymmetric.  It is also remarkable that metrics with antisymmetric
part are exactly what are needed in string theory to establish
T-duality. In summary, one has on the table a general
noncommutative Riemannian geometry to play with. It can be applied to
a variety of algebras far removed from conventional geometry. Some finite
dimensional examples will be presented elsewhere.

\subsection*{Acknowledgements} It is a pleasure to thank the
organisers of the LMS symposium in Durham for a thoroughly
enjoyable conference.

config: arxiv


TITLE: Derivative of trace and determinant of matrix with respect to a vector
QUESTION [2 upvotes]: I encounter a difficulty in one of my research dealing with derivative of a matrix. I will appreciate much for your expertise on this.
Let $\mathbf{x }=vec\left( X \right) $ where $X $ is a $T\times r
$ matrix of unknown parameters ($T$ and $r$ are constant, $T>>r$) and $vec$
is the vectorization, let
\begin{equation}
\mathbf{H}\left( \mathbf{X }\right) =\left( X ,\mathbf{I}_{T}\right) ^{\prime },  \label{def_H_theta}
\end{equation}
where $\mathbf{I}_T$ is $T \times T $
identity matrix. Let
\begin{equation}
\Omega \left( \mathbf{X }\right) _{T\times T}=\left( \mathbf{I}%
_{T}+XX ^{\prime }\right) .  \label{var-cov_TS_multi}
\end{equation}
What is the derivative of
\begin{equation}
\frac{\partial \log \left\vert \Omega \left( \mathbf{X }\right)
\right\vert }{\partial \mathbf{x }}=??
\end{equation}
$|A|$ is the determinant of $A$,
and
\begin{equation}
\frac{\partial tr\left[ \mathbf{H}\left( \mathbf{X }\right) ^{\prime
}\Omega \left( \mathbf{X }\right) ^{-1}\mathbf{H}\left( \mathbf{%
X }\right) \right] }{\partial \mathbf{x }}=??
\end{equation}
where $tr$ is the trace operator.
Many thanks!

REPLY [2 votes]: For typing convenience, let me substitute Latin in place of your Greek letters
$$\eqalign{
 X &= \Xi \cr
 x &= \vartheta = \operatorname{vec}(X) \cr
 W &= \Omega = I + XX^T = W^T \cr\cr
}$$
Then for your first function, the differential and gradient can be calculated as
$$\eqalign{
 f &= \log\det W \cr
\cr
df &= d\log\det W = d\operatorname{tr}\log W \cr
   &= W^{-T}:dW \cr
   &= W^{-T}:(dX\,X^T+X\,dX^T) \cr
   &= \big(W^{-T}X+W^{-1}X\big):dX \cr
   &= 2\,W^{-1}X:dX \cr
   &= 2\,\operatorname{vec}(W^{-1}X)\cdot\,dx \cr
\cr
\frac{\partial f}{\partial x} &= 2\,\operatorname{vec}(W^{-1}X) \cr\cr
}$$
In your second function, I don't quite understand the definition of $\,H(\vartheta)\,$ so all I can offer is a partial solution
$$\eqalign{
 M &= W^{-1} = M^T \cr
 f &= H:MH \cr
\cr
df &= dH:MH + H:M\,dH + H:dM\,H \cr
   &= (M+M^T)H:dH + HH^T:dM \cr
   &= 2\,MH:dH - HH^T:M\,dW\,M \cr
   &= 2\,MH:dH + MHH^TM:dW \cr
   &= 2\,MH:dH + MHH^TM:(dX\,X^T+X\,dX^T) \cr
   &= 2\,MH:dH + 2\,MHH^TMX:dX \cr
\cr
}$$
You can finish off the solution by expanding $dH$ in terms of $dX$, then vectorizing.
In the above, a colon denotes the double-dot (aka Frobenius) product, which is merely a product notation for the trace, i.e. $$A:B=\operatorname{tr}(A^TB)$$
Update
I'm guessing that $H$ is a partitioned matrix: $H=[\,X, I\,]$
Expanding that term in the differential
$$\eqalign{
 MH:dH &= [\,MX, M\,]:[\,dX, 0\,] \cr
 &= MX:dX + M:0 \cr
}$$
So, continuing with the full differential
$$\eqalign{
 df &= 2\,\Big(MX + MHH^TMX\Big):dX \cr
   &= 2\,\operatorname{vec}(MX + MHH^TMX)\cdot dx \cr
\cr
\frac{\partial f}{\partial x} &= 2\,\operatorname{vec}(MX + MHH^TMX) \cr
}$$


TITLE: Projections of sets in product sigma algebras over a countable set
QUESTION [1 upvotes]: I've been looking through some posts about how projections of sets from product $\sigma$-algebras of two Borel $\sigma$-algebras onto their component spaces are not necessarily measurable which apparently leads into the rabbit hole of descriptive set theory.
My problem, though, is a little different and I thought perhaps someone had an idea of how to tackle it seeing as how I am not really knowledgeable when it comes to descriptive set theory.
What I have is a kind of Polish space $\mathcal{X}$ (complete, separable metric  space), a countable (or even finite) set $\mathcal{A}$ (being polish by virtue of its discrete metric if I am not mistaken) and their respective Borel $\sigma$-algebras $\mathcal{B}({\mathcal{X}})$ and $\mathcal{B}(\mathcal{A})=2^{\mathcal{A}}$ since $\mathcal{A}$ is countable.
What I am wondering now is, if I look at some set from their product $S \in \mathcal{B}({\mathcal{X}}) \otimes \mathcal{B}(\mathcal{A})$, does the setup now give us that
$$ \pi_{\mathcal{X}}(S) := \{ x \in \mathcal{X} | \, \exists\, a \in \mathcal{A}: \, (x,a) \in S \} $$
What I have tried to use is that since $\mathcal{B}(\mathcal{A})=2^{\mathcal{A}}$ and countable, we can write $S$ as
$$ S = \bigcup_{a \in \mathcal{A}} S_a \times \{a \} $$
where $ S_a = \{ x \in \mathcal{X}|\, (x,a) \in S \} $ hoping I could somehow conclude that $ S_a \in \mathcal{B}(\mathcal{X}) $ as intuitively, pairing Borel sets with countably (or even finitely) many point sets and using that as a generator for $ \mathcal{B}({\mathcal{X}}) \otimes \mathcal{B}(\mathcal{A}) $ shouldn't all of a sudden provide us with sets whose projections are non-Borel.
However, I cannot seem to properly prove this statement and perhaps it is false to begin with. I thought someone here could give some insights.
Thanks!

REPLY [0 votes]: Your argument goes through. The set $S_a = \{ x \in \mathcal{X}|\, (x,a) \in S \}$ is known as the $a$-section, and it is generally measurable for the product $\sigma$-algebra (the smallest $\sigma$-algebra generated by rectangles).
The proof that sections are measurable is a crucial step in the proof of Fubini's theorem, and that is where you can probably find a proof in most textbooks that treat measure theory. You can also find a short proof here.


TITLE: Counting points in/on cuboid
QUESTION [0 upvotes]: Given a cuboid that extend in x,y,z axis such that |x|≤N, |y|≤N, |z|≤N where N is given and can have value up to 10^9.Now a shooter is standing at origin (0,0,0).He need to shoot on any of the surfaces of the cuboid in such a way that these 2 conditions are satisified :

Their are atleast M integeral points (1≤M≤1000) on the line joining (0,0,0) and the point on surface (x,y,z) where shooter points.
Also count of these integral points should be divisble by given number D where 1≤D≤1000

Now we need to count the number of such lines along which the shooter can point his gun in such a way that these 2 conditions are satisified.
Example :
Let N=3 , M=2 and D=1 then here answer will be 26.
The directions in which shooter can point to satisfy these conditions are : (−1,−1,−1), (−1,−1,0), (−1,−1,1), (−1,0,−1), (−1,0,0), (−1,0,1)
(−1,1,−1), (−1,1,0), (−1,1,1), (0,−1,−1), (0,−1,0), (0,−1,1)
(0,0,−1), (0,0,1), (0,1,−1), (0,1,0), (0,1,1), (1,−1,−1)
(1,−1,0), (1,−1,1), (1,0,−1), (1,0,0), (1,0,1), (1,1,−1)
(1,1,0), (1,1,1)

REPLY [0 votes]: HINT:
Suppose the shooter shoots at $(X,Y,Z)$, then the line joining this and origin will be $$ x=X/t,y=Y/t,z=Z/t;t>1$$
Suppose I take the point $(100,200,300)$ on a surface, then possible integral points are:
$$(1,2,3),(2,4,6),(3,6,9),(4,8,12),...$$
So actually you have to select three mutually co-prime numbers and make multiples of them until you reach the border.Any ideas?
Now count of these point must be divisible by $D$. Any ideas (think in term of number of factors)?


TITLE: How prove this $\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$
QUESTION [13 upvotes]: Given the positive integer numbers $n$,prove that
$$\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$$
where $\sigma(n)$ is defined as $$\sigma{(N)}=\sum_{d|N}d$$
This problem I think is very interesting, and have nice methods?
Thank you everyone.
and I find some of this

or can see:http://www.advancesindifferenceequations.com/content/pdf/1687-1847-2013-84.pdf

REPLY [7 votes]: It is difficult to add anything to such a nice answer. So take this more as a comment, too long for the the comment field:
There is another elementary proof, using only
an arithmetic identity, which generalizes a classical formula of Liouville. For details, see the article "Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions" by Huard et al.On page $21$ there is Theorem $5$, which yields
$$
\sum_{k\le N/2} \sigma(2k-1)\sigma(N-2k+1)=\frac{1}{24}\left( 5\sigma_3(N)-21\sigma_3(N/2)+16\sigma_3(N/4)\right)+(1 − 6N)(σ(N) − 3σ(N/2) + 2σ(N/4))).
$$
For $N=2^n$ there result follows also (with some manipulations given in theorem $9$). The proof is completely elementary, only using the generalization of Liouville’s Formula, which is Theorem $1$.


TITLE: Find sum of roots of complex equation $z^6 = z^{18} = -1$
QUESTION [4 upvotes]: Find sum of roots of equation $z^6 = z^{18} = -1$
Roots of $z^6=-1$

Roots of $z^{18}=-1$

From the circle I can see, that:
$\{z : z^6=-1\} \cap \{z : z^{18}=-1\} = \{z : z^6=-1\}$
So, I can notice, that sum of roots $z^6=-1$ is equal to zero.
However, I solved that with help of WolframAlpha - drawing a circle with 18 points (or calculating all roots) is not very encouraging. Is there the more reliable method to solve such problem?
Edit: correct circles are:

REPLY [1 votes]: If $\alpha$ is a root, $-\alpha$ is a different root. Both sum up to $0$


TITLE: Riesz representation theorem in a multivariate setting
QUESTION [1 upvotes]: Riesz representation theorem says that if $H$ is a Hilbert space, and $f\in H^*$, then there exists a unique $y \in H$ such that $f(x)=\langle y,x \rangle_H$.
Now suppose we look at $H\times H$. This is still a Hilbert space, equipped with the scalar product $\langle (x_1,x_2),(y_1,y_2)\rangle_{H \times H}= \langle x_1,y_1 \rangle_H + \langle x_2,y_2 \rangle_H$
Clearly we can apply Riesz representation theorem to $H\times H$: $\forall f \in (H\times H)^*$, there exist a unique $(y_1,y_2) \in H \times H$ such that $f(x_1,x_2)= \langle (y_1,y_2), (x_1,x_2)\rangle_{H\times H}$
The (unproved) result in my book says : by Riesz representation theorem, $\forall f :H\times H \rightarrow \mathbb R $ bilinear and continuous, there exist a unique $Q \in L(H)$ such that $f(x_1,x_2)= \langle Qx_1, x_2\rangle_{H}$, where $L(H)$ is the set of bounded linear operators $T:H \rightarrow H$.
How do we prove that ? Do we have to use Riesz on $H\times H$ or simply on $H$ ?

REPLY [2 votes]: This is not true as stated: the function $(x_1,x_2) \mapsto \langle Qx_1,x_2\rangle$ is not  a linear mapping from $H\times H$ to $\mathbb R$. Rather it is bilinear.
Please check the assumptions on $f$ again.


TITLE: Let $a_{1} = 1$ and $a_{n} = n(a_{n-1}+1)\;\forall n\geq 2.$ Then $ \lim_{n\rightarrow \infty} \prod^{n}_{r=1}\left(1+\frac{1}{a_{r}}\right)$
QUESTION [1 upvotes]: Let $a_{1} = 1$ and $a_{n} = n(a_{n-1}+1)\;\forall n\geq 2.$ Then $\displaystyle \lim_{n\rightarrow \infty} \prod^{n}_{r=1}\left(1+\frac{1}{a_{r}}\right)$

$\bf{My\; Try::}$ Given  $a_{n} = n(a_{n-1}+1)$
Replace $n\rightarrow n+1\;,$ We get $$a_{n+1}=(n+1)(a_{n}+1)\Rightarrow \frac{a_{n+1}}{n+1} = a_{n}+1$$
Now how can i solve it, Help required, Thanks

REPLY [0 votes]: Note that
$$
\frac{a_n}{n!}=\frac{a_{n-1}}{(n-1)!}+\frac1{(n-1)!}=1+1+\frac1{2!}+…+\frac1{(n-1)!}
$$
and as SchrodingersCat demonstrated, the finite products are
$$
\prod_{r=1}^n\left(1+\frac1{a_r}\right)=\frac{a_{n+1}}{(n+1)!}
$$
which now has an obvious limit.


TITLE: Proof - expectation of a discrete random variable
QUESTION [1 upvotes]: A discrete random variable $X >0$
Can you prove : $$\mathbb{E}[X] = \sum_{x>0} \mathbb{P}[ X >= x ]$$
I don't get the question - is it asking for a generic proof or to find an $x$ which satisfies?
Thanks.

REPLY [2 votes]: The right statement should be $EX=\sum_{x=1}^\infty P(X\geq x)$, the index $x$ must belong to $\mathbb N$, not $\mathbb R_+$.
We have
\begin{align*}
\sum_{x=1}^\infty P(X\geq x) &=\sum_{x=1}^\infty \sum_{n=x}^\infty P(X=n)\\
&=\sum_{n=1}^\infty \sum_{x=1}^n P(X=n)\\
&=\sum_{n=1}^\infty nP(X=n)=EX.
\end{align*}