text:
\section{Method Description} \label{sec:method}

In this section we present the detail of our method for deep
learning of non-autonomous systems \eqref{eq:ode}. The key ingredients
of the method include: (1) parameterizing
the external input $\gamma(t)$ locally (in time); (2) decomposing the
dynamical system into a modified system comprising of a sequence of
local systems; and (3) deep learning of the local systems.

\subsection{Local Parameterization}

The analytical solution of the unknown system \eqref{eq:ode} satisfies
$$
\x(t) = \x_0 + \int_0^t \f(\x(s), \gamma(s)) ds.
$$
Our learning method aims at providing accurate approximation to the true solution
at a prescribed set of discrete time instances,
\be \label{tline}
0= t_0 < t_1 <\cdots < t_n <\cdots <t_N = T,
\ee
where $T>0$. Let
$$
\delta_n = t_{n+1} - t_n, \qquad n=0,\dots, N-1,
$$
be the time steps, the exact solution
satisfies, for $n=0,\dots, N-1$,
\begin{equation} \label{evo}
  \begin{split}
  \x(t_{n+1}) &= \x(t_n) + \int_{t_n}^{t_{n+1}} \f(\x(s), \gamma(s)) ds
  \\
  & = \x(t_n) + \int_{0}^{\delta_{n}} \f(\x(t_n+\tau),
  \gamma(t_n+\tau)) d\tau.
  \end{split}
\end{equation}

For each time interval $[t_n, t_{n+1}]$, $n=0,\dots, N-1$, we first
seek a local parameterization for the external input function
$\gamma(t)$, in the following form,
\begin{equation}
\label{local_param}
\tgamma_n(\tau;\ggamma_n) := \sum_{j=1}^{n_b} \hgamma_n^j
    b_j(\tau)\approx \gamma(t_n+\tau), \qquad \tau\in [0, \delta_{n}],
\end{equation}
where $\{b_j(\tau), j=1,\dots, n_b\}$ is a set of prescribed analytical basis functions and
\be \label{Gamma}
\ggamma_n=(\hgamma_n^1, \dots, \hgamma_n^{n_b})\in \Rs^{n_b}
\ee
are the basis coefficients  parameterizing the local input $\gamma(t)$ in $[t_n, t_{n+1}]$.

Note that in many practical applications, the external input/control
process $\gamma(t)$ is already prescribed in a parameterized
form. In this case, the local parameterization \eqref{local_param}
becomes exact, i.e., $\gamma(t_n+\tau) = \tgamma_n(\tau;\ggamma_n,)$.
In other applications when the external input
$\gamma(t)$ is only known/measured at certain time instances, a
numerical procedure is required to create the parameterized form
\eqref{local_param}. This can be typically accomplished via a numerical
approximation method, for example, Taylor expansion, polynomial
interpolation, least squares regression etc.

\subsection{Modified System}

With the local parameterization \eqref{local_param} constructed for
each time interval $[t_n, t_{n+1}]$, we proceed to define a global
parameterized input
\be \label{global_gamma}
\tgamma(t; \ggamma)=\sum_{n=0}^{N-1}
\tgamma_n(t-t_n;\ggamma_n)\mathbb{I}_{[t_n, t_{n+1}]}(t),
\ee
where
\be \label{global_para}
\ggamma =
\{\ggamma_n\}_{n=0}^{N-1}\in\Rs^{N\times n_b}
\ee
is global parameter set for $\tgamma(t)$, and
$\mathbb{I}_A$ is indicator function satisfying, for a set $A$,
$\mathbb{I}_A(x) =1$ if $x\in A$ and 0 otherwise.

We now define a modified system, corresponding
to the true (unknown) system \eqref{eq:ode}, as follows,
\begin{equation}
  \label{eq:ode_mod}
  \left\{
\begin{split}
&\frac{d}{dt}\tx (t)=\f(\tx, \tgamma(t; \ggamma)),\\
&\tx(0)=\x_0,
\end{split}
\right.
\end{equation}
where $\tgamma(t; \ggamma)$ is the globally parameterized input defined in \eqref{global_gamma}.
Note
that when the system input $\gamma(t)$ is already known or given in a parametric
form, i.e. $\tgamma(t) = \gamma(t)$, the modified system
\eqref{eq:ode_mod} is equivalent to the original system \eqref{eq:ode}.
When the parameterized process $\tgamma(t)$
needs to be numerically constructed, the modified system \eqref{eq:ode_mod}
becomes an approximation to the true system \eqref{eq:ode}. The
approximation accuracy obviously depends on the accuracy in
$\tgamma(t)\approx \gamma(t)$.
For the modified system, the following results holds.

\begin{lemma}
	\label{thm:theorem1}
Consider system \eqref{eq:ode_mod} over the discrete set of time
instances \eqref{tline}.
There exists a function $\tpphi: \R^d\times
\R^{n_b}\times \R\rightarrow \mathbb{R}^d$, which depends on $\f$, such that
for any time interval $[t_n, t_{n+1}]$, the solution of
\eqref{eq:ode_mod} satisfies
\begin{equation}
	\label{evo_mod}
	\tx(t_{n+1})=\tx(t_n)+	\tpphi(\tx(t_n), \ggamma_n, \delta_n),
        \qquad n= 0, \dots, N-1,
	\end{equation}
	where $\delta_n=t_{n+1}-t_n$ and $\ggamma_n$ is the local
        parameter set \eqref{Gamma} for the locally parameterized input $\tgamma_n(t)$ \eqref{local_param}.
\end{lemma}
\begin{proof}
 Let $\tx_n(t) $ denote $\tx(t)$ in the time interval $[t_n,
 t_{n+1}]$, i.e.,
 $$
 \tx(t) = \sum_{n=0}^{N-1}
\tx_n(t)\mathbb{I}_{[t_n, t_{n+1}]}(t).
$$
  With the global input $\tgamma(t)$ defined in the piecewise manner
  in \eqref{global_gamma}, the system \eqref{eq:ode_mod} can be
  written equivalently as,
  for each interval $[t_n, t_{n+1}]$, $n=0,\dots, N-1$,
  \begin{equation*}
  \left\{
    \begin{split}
  &\frac{d}{dt}\tx_n (t)=\f(\tx_n, \tgamma_n(t-t_n; \ggamma_n)), \qquad
  t\in (t_n, t_{n+1}], \\
  & \tx_n(t_n) = \tx(t_n).
\end{split}
\right.
\end{equation*}
Let $\PPhi_n: (\R^d\times\R)\times\R\to \R^d$ be its (time dependent)
flow map such that
$$
\tx_n(r) = \PPhi_n((\tx_n(s),s),r-s), \qquad t_n\leq s\leq r\leq t_{n+1}.
$$
We then have
\be \label{tx}
\tx_n(t_n+\tau) = \PPhi_n((\tx(t_n),0), \tau), \qquad \tau\in
[0,\delta_n],
\ee
where the initial condition $\tx_n(t_n) = \tx(t_n)$ has been used.

The solution of \eqref{eq:ode_mod} from $t_n$ to $t_{n+1}$ satisfies
	\begin{align*}
	\tx(t_{n+1})&=\tx(t_n)+\int_{t_n}^{t_{n+1}} \f(\tx(t),
                      \tgamma(t; \ggamma)) dt\\

          &=\tx(t_n)+\int_{0}^{\delta_{n}} \f(\tx_n(t_n+\tau),
            \tgamma_n(\tau; \ggamma_n))d\tau\\
          &=\tx(t_n)+\int_{0}^{\delta_{n}} \f(\PPhi_n((\tx(t_n),0), \tau), \tgamma_n(\tau; \ggamma_n))d\tau,

	\end{align*}
        where \eqref{global_gamma} and \eqref{tx} have been
        applied. Let
	\begin{equation*}
	\tpphi(\tx(t_n), \ggamma_n, \delta_n) :=\int_{0}^{\delta_{n}}
        \f(\PPhi_n((\tx(t_n),0), \tau), \tgamma_n(\tau;
        \ggamma_n))d\tau
      \end{equation*}
and the proof is complete.
   \end{proof}

\subsection{Learning of Modified Systems}

The function $\tpphi$ in \eqref{evo_mod} governs the evolution of the
solution of the modified system \eqref{eq:ode_mod} and
is the target function for our proposed deep learning method. Note
that in each time interval $[t_n, t_{n+1}]$ over the prediction time
domain \eqref{tline}, the solution at $t_{n+1}$ is determined by its
state at $t_n$, the local parameter set $\ggamma_n$ for the local
input $\tgamma_n$, the step size $\delta_n = t_{n+1}-t_n$, and
obviously, the
form of the original equation $\f$. Our learning algorithm thus seeks
to establish and train a deep neural network with input $\tx(t_n)$,
$\ggamma_n$, $\delta_n$ and
output $\tx(t_{n+1})$. The internal feed-forward network connecting the
input and output thus serves as a model of the unknown dynamical
system \eqref{eq:ode}.

\subsubsection{Training Data Set}

To construct the training data set, we first re-organize the original data
set \eqref{data_set}. Let us assume the length of each trajectory data in
\eqref{data_set} is at least 2, i.e., $K^{(i)}\geq 2$, $\forall i$. We
then re-organize the data into pairs of two adjacent time instances,
\begin{equation} \label{data_pair}
\left\{ \x\left(t^{(i)}_k\right), \x\left(t^{(i)}_{k+1}\right); \gamma^{(i)}\right\}, \qquad k=1,\dots, K^{(i)}-1,
 \quad i=1,\dots, N_T,
\end{equation}
where $N_T$ is the total number of data trajectories. Note that for
each $i=1,\dots, N_T$, its trajectory is driven by a known external input
$\gamma^{(i)}$, as shown in \eqref{data_set}.  We then seek, for
the time interval $[t_k^{(i)}, t_{k+1}^{(i)}]$ with $\delta_k^{(i)}
  =t_{k+1}^{(i)}- t_{k}^{(i)}$, its local
parameterized form $\tgamma_k^{(i)}(\tau; \ggamma_k^{(i)})$, where
$\tau\in [0, \delta_k^{(i)}]$ and $\ggamma_k^{(i)}$ is the parameter
set for the local parameterization of the input, in the form of
\eqref{local_param}.  Again, if the external
input is already known in an analytical parametric form, this step is
trivial; if not this step usually requires a standard
regression/approximation procedure and is not discussed in detail here
for the brevity of the paper.

For each data pair \eqref{data_pair}, we now have its associated time
step $\delta_k^{(i)}$ and local parameter set $\ggamma_k^{(i)}$ for
the external input. The total number of such pairings is
$K_{tot}= K^{(1)}+K^{(2)}+\cdots K^{(N_T)}-N_T$.
We then proceed to select $J\leq K_{tot}$ number of such pairings to
construct the training data set for the neural network model. Upon
re-ordering using a single index, the training data set takes the
following form
\be \label{training_set}
\mathcal{S} = \left\{(\x^{(j)}_k, \x^{(j)}_{k+1}); \ggamma^{(j)}_k,
  \delta^{(j)}_k\right\}, \qquad j=1,\dots, J,
\ee
where the superscript $j$ denotes the $j$-th data entry, which belongs
a certain $i$-th trajectory in the original data pairings
\eqref{data_pair}. The re-ordering can be readily enforced to be
one-on-one, with the trajectory information is implicitly embedded. Note
that one can naturally select all the data pairs in \eqref{data_pair} into the training
data set \eqref{training_set}, i.e., $J=K_{tot}$. In practice, one may
also choose a selective subset of  \eqref{data_pair} to construct the
training set \eqref{training_set}, i.e.. $J<K_{tot}$, depending on the
property and quality of the original data.

\subsubsection{Network Structure and Training}

With the training data set \eqref{training_set} available, we proceed
to define and train our neural network model. The network model seeks
to learn the one-step evolution of the modified system, in the form
of \eqref{evo_mod}. Our proposed network model defines a
mapping $\tN: \R^{d+n_b+1}\to\R^d$, such that
\be
\X_{out} = \tN(\X_{in}; \Theta), \qquad \X_{in} \in \R^{d+n_b+1},
\quad \X_{out} \in \R^d,
\ee
where $\Theta$ are the network parameters that need to be trained.
The network structure is
illustrated in Fig.~\ref{fig:Net}.

\begin{figure}
	\centering
	\includegraphics[scale=0.4]{./Figures/DNN.pdf}
	\caption{Illustration of the proposed neural network.}
	\label{fig:Net}
      \end{figure}

Inside the network, $\N:\R^{d+n_b+1}\to \R^{d}$ denotes the operator
associated with a feed-forward neural network with $(d+n_b+1)$ input
nodes and $d$ output nodes.  The input is multiplied with $\hI$ and
then re-introduced back before the final output. The
operator $\hI\in \R^{d\times (d+n_b+1)}$ is a matrix of size ${d\times
  (d+n_b+1)}$. It takes the form
\be \label{hatI}
\hI = [ \I_d, \mathbf{0} ],
\ee
where $\I_d$ is identity matrix of size $d\times d$ and $\mathbf{0}$
is a zero matrix of size $d\times (n_b+1)$. Therefore, the network
effectively defines a mapping
\be \label{net_model}
\X_{out} = \tN(\X_{in}; \Theta) = [\hI + \N(\cdot; \Theta)](\X_{in}).
\ee

Training of the network is accomplished by using the training data set
\eqref{training_set}. For each of the $j$-th data entry, $j=1,\dots,
J$, we set
\be \label{X_in}
\X_{in}^{(j)} \leftarrow [\x_k^{(j)}; \ggamma_k^{(j)}; \delta_k^{(j)}] \in
\R^{d+n_b+1}.
\ee
The network training is then conducted by
minimizing the mean squared loss between the network output $\X_{out}^{(j)}$
and the data $\x_{k+1}^{(j)}$, i.e.,
\be
\label{eq:loss}
\Theta^*=\argmin_{\Theta} \frac{1}{J}\sum_{\j=1}^J \left\|\tN(\X_{in}^{(j)}; \Theta)-\x^{(j)}_{k+1}\right\|^2.
\ee

\subsubsection{Learned Model and System Prediction}

Upon satisfactory training of the network parameter using
\eqref{eq:loss}, we obtain a trained network model for the unknown
modified system \eqref{eq:ode_mod}
\be \label{trained}
\X_{out} = \tN(\X_{in}; \Theta^*) = [\hI + \N(\cdot; \Theta^*)](\X_{in}),
\ee
where $\hI$ is defined in
\eqref{hatI} and $\N$ is the operator of the FNN,
as illustrated in the
previous section and in Fig.~\ref{fig:Net}.

For system prediction with a given external input function
$\gamma(t)$, which is usually not in the training data set, let us consider the time instances
\eqref{tline}. Let
$$
\X_{in} = [\x(t_n); \ggamma_n; \delta_n]
$$
be a concatenated vector consisting of
the state
variable at $t_n$, the parameter vector for the local parameterization of
the external input between $[t_n, t_{n+1}]$, and $\delta_n =
t_{n+1}-t_n$. Then, the trained model produces a one-step evolution of
the solution
\be
\label{evo_mod_new}
\hx(t_{n+1})= \x(t_n) + \N(\x(t_n), \ggamma_n, \delta_n; \Theta^*).
\ee

Upon applying \eqref{evo_mod_new} recursively, we obtain a network model for predicting the system states of the
unknown non-autonomous system \eqref{eq:ode}. For a given initial
condition $\x_0$ and external input $\gamma(t)$,
\begin{equation}
\label{eq:prediction}
\left\{
\begin{split}
&\wh{\x}(t_0) = \x_0, \\
&\wh{\x}(t_{n+1}) = \wh{\x}(t_{n}) + {\N}(\wh{\x}(t_n), \ggamma_n, \delta_n; \Theta^*), \\
&t_{n+1} =
t_n + \delta_n, \qquad n=0,\dots, N-1,
\end{split}
\right.
\end{equation}
where $\ggamma_n$ are the parameters in the local parameterization of
$\gamma(t)$ in the time interval $[t_n, t_{n+1}]$.

It is obvious that the network predicting model \eqref{evo_mod_new}
is an approximation to the one-step evolution \eqref{evo_mod} of the modified
system \eqref{eq:ode_mod}, which in turn is an approximation of the original
unknown dynamical system \eqref{eq:ode}. Therefore,
\eqref{eq:prediction} generates an approximation to the solution of
the unknown system \eqref{eq:ode} at the discrete time instances
$\{t_n\}$ \eqref{tline}.


TITLE: Equivalent formulation of complexity theory in Lambda Calculus?
QUESTION [11 upvotes]: In complexity theory the definition of time and space complexity both reference a universal Turing machine: resp. the number of steps before halting, and the number of cells on the tape touched.
Given the Church-Turing thesis, it should be possible to define complexity in terms of lambda calculus as well.
My intuitive notion is that time complexity can be expressed as the number of β-reductions (we can define away α-conversion by using De Brujin indexes, and η is barely a reduction anyway), while space complexity can be defined as the number of symbols (λ's, DB-indexes, “apply”-symbols) in the largest reduction.
Is this correct? If so, where can I get a reference? If not, how am I mistaken?

REPLY [5 votes]: A note about space complexity.  While, as pointed out by Martin in his answer, the naive way to count time complexity turns out to work well, the definition of space complexity you suggest is easily seen to be inadequate.  Indeed, in the case of space you really want to be able to speak of sublinear complexity, e.g. you want to be able to recover the class $\mathsf L$ (deterministic logspace, which is to space a bit what $\mathsf P$ is to time), and your definition obviously does not allow you to do that: in any reduction $M\to^\ast N$, your are counting at least the size of $M$, which is linear in the input size.
The moral of the story is that rewriting is not suitable for counting space.  Ulrich Schöpp and Ugo Dal Lago were the first to advocate the use of the so-called geometry of interaction (GoI) for dealing with sublinear space complexity (cf. their ESOP 2010 paper "Functional Programming in Sublinear Space").  As far as I know, the GoI is used in one way or another in all lambda-calculus-based characterizations of sublinear space classes.  I do not want to get into what the GoI is here; let's say that it is a way of executing a lambda-term without reducing it (i.e., without firing $\beta$-redexes) but by "traveling" through its syntactic tree with a certain auxiliary information.


TITLE: Loudspeaker attached to cylinder - closed or open?
QUESTION [0 upvotes]: I have a loudspeaker that is facing into a long cylindrical piece of PVC pipe of length L. The loudspeaker is sealed in so that no air can escape around the edges. The pipe is open at the other end.
The loudspeaker is attached to a signal generator and made to vibrate at frequency f.
In this configuration, should the system be modeled as a closed pipe or an open pipe? Or does the choice of model depend on f ?
My intuition is that the loudspeaker represents a displacement antinode (and therefore a pressure node), and so the system should be modeled as an open pipe (pressure nodes on both ends). The system would then resonate at frequencies where $L=\frac{\lambda}{2}=\frac{c}{2f}$. Is this a correct assumption?
If one were to place a circular piece of PVC with a small hole in the center across the face of the speaker, so that air is compressed inside a small cavity before being released into the pipe, would this reverse the scenario so that the speaker end becomes a pressure antinode?
Thanks in advance!

REPLY [1 votes]: I decided to test this using approx 1m long PVC pipe from Bunnings and a small loudspeaker.
I had two pieces of light milk-carton plastic which I placed over the front of the loudspeaker - one with a small hole in the center, and one with a large hole. Then for each of the pieces of plastic, I placed the PVC pipe on top, pressing down the plastic between the loudspeaker and the pipe to seal it.
Using the plastic with a large hole, the lowest loud resonance was at 166.5Hz, which corresponds to a wavelength of 2.06m. This means that the 1m pipe was generating half-wavelength standing waves, and therefore the system should be modeled as an open cylinder with pressure nodes at each end. This contradicts Niels Nielson's answer (sorry).
Using the plastic with a small hole, the lowest loud resonance was one octave lower at 83.25Hz, which corresponds to a wavelength of 4.12m. This means that the 1m pipe was generating quarter-wavelength standing waves, and therefore the system should be modeled as an closed cylinder with a pressure node at the open end and a pressure antinode at the speaker end.
So my intuition was correct!
It is possible that I bungled the experiment so if anyone has anything to add I would love to hear it.
Cheers


TITLE: Semantics for minimal logic
QUESTION [16 upvotes]: Minimal logic is a fragment of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet). Essentially, from proof-theoretical viewpoint, this means that in minimal logic the bottom $\bot$ is considered as a propositional variable, without any special inference rule involving it.
Question: Is there a semantics $\mathcal{S}$ for minimal logic such that a completeness theorem holds? By completeness theorem I mean a statement of the form (for both propositional and first-order languages):

For every formula $A$, $A$ is provable in minimal logic if and only if $A$ is valid in all $\mathcal{S}$-structures.

I guess such a semantics could be a generalization of intuitionistic Kripke models.
I would like also to have some references about completeness theorem in (propositional and/or first-order) minimal logic.

REPLY [8 votes]: Yes, tweaking Kripke semantics based on that for intuitionistic logic does the trick. If I recall right, there are slightly different ways of doing this.
Here's a version for minimal logic and some variants, in a recent paper by Almudena Colacito, Dick de Jongh and Ana Lucia Vargas.


TITLE: For Which Value The Matrix is Diagonalizable?
QUESTION [2 upvotes]: For which values of $a$ the matrix $\left(\begin{array}{ccc} 2 & 0 & 0 \\ 2 & 2 & a \\ 2 & 2 & 2 \end{array}\right)$ is diagonalizable:

above $\mathbb{R}$
above $\mathbb{C}$

We need to look at the characteristic polynomial which is $(x-2)^3-2a(x-2)=x^3-6x^2+2x(6-a)+4(a-2)$ , How do I find the eigenvalues one is $2$ and the other? and how so I continue?

REPLY [0 votes]: $x^3-6x^2+2x(6-a)+4(a-2)$. Clearly it shows that  $x=2$ is a zero of the polynomial. Now ,
$x^3-6x^2+2x(6-a)+4(a-2)=0$
$\implies (x-2)(x^2-4x+4-2a)=0$
$\implies x=2 , x=\frac{4\pm\sqrt{16-4(4-2a)}}{2}=2\pm\sqrt{2a}.$
If $a=0$ then , all roots are $2$ and then the matrix is NOT diagonalizable(check!)
If $a\not=2$ then the matrix is diagonalizable.


text:
\begin{document}

\maketitle

\begin{abstract}
In this article, the existence of Ulrich bundles on projective bundles $\bP(\cE) \to X$ is discussed.
In the case, that the base variety $X$ is a curve or surface, a close relationship between Ulrich bundles on $X$ and those on $\bP(\cE)$ is established for specific polarisations.
This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and some surfaces.
\end{abstract}

\section{Introduction}

Given a smooth projective variety $X$, polarised by a very ample divisor $A$,
let $i \colon X \into \bP^N$ be the associated closed embedding.
A locally free sheaf $\cF$ on $X$ is called \emph{Ulrich bundle} (with respect to $A$) if and only if it satisfies one of the following conditions:
\begin{itemize}
\item There is a linear resolution of $\cF$:
\[
0 \to \cO_{\bP^N}(-c)^{\oplus b_c} \to  \cO_{\bP^N}(-c+1)^{\oplus b_{c-1}} \to \cdots \to  \cO_{\bP^N}^{\oplus b_0} \to i_* \cF \to 0,
\]
where $c$ is the codimension of $X$ in $\bP^N$.
\item The cohomology $\sH^\bullet(X, \cF(-pA))$ vanishes for $1 \leq p \leq \dim(X)$.
\item For any finite linear projection $\pi \colon X \to \bP^{\dim(X)}$, the locally free sheaf $\pi_* \cF$ splits into a direct sum of $\cO_{\bP^{\dim(X)}}$.
\end{itemize}
Actually, by \cite{Eisenbud-Schreyer-Weyman}, these three conditions are equivalent.
One guiding question about Ulrich bundles is whether a given variety admits an Ulrich bundle of low rank.
The existence of such a locally free sheaf has surprisingly strong implications about the geometry of the variety, see the excellent surveys \cite{Beauville-Intro, Coskun-Intro}.

Given a projective bundle $\pi \colon \bP(\cE) \to X$, this article deals with the question, what is the relation between Ulrich bundles on the base $X$ and those on $\bP(\cE)$?
Note that answers to such a question depend much on the choice of a very ample divisor.
Here we consider very ample divisors $D = \pi^* A + H$, where $H$ is the relative hyperplane section of $\pi \colon \bP(\cE) \to X$.
In this case, we ask when a locally free sheaf of the form $\pi^* \cF (D)$ is an Ulrich bundle with respect to $D$.
By \autoref{prop:bundles-general}, this can be expressed solely in terms of cohomology vanishing on the base $X$.
If the base variety is a curve, then, essentially as a corollary, we obtain the following statement, see \autoref{thm:bundles-curves} and \autoref{rem:bundles-curves}.

\begin{theoremalpha}
\label{main:curve}
Let $\pi \colon \bP(\cE) \to C$ be a projective bundle over a smooth projective curve,
and let $D = \pi^* A + H$ be very ample.
Then a locally free sheaf $\pi^* \cF (D)$ is Ulrich if and only if $\sH^\bullet(C,\cF)=0$.

In particular, there are Ulrich line bundles on $\bP(\cE)$ with respect to $D$.
\end{theoremalpha}

Note that, for any very ample divisor $A'$ on $C$, the locally free sheaf $\cF$ gives rise to the Ulrich bundle $\cF(A')$ on $C$.
This result is a generalisation of \cite[Thm. 2.1]{Aprodu-etal}, where the existence of Ulrich line bundles on $\bP^1$-bundles over curves is shown.

If the base variety is a surface, then the connection between Ulrich bundles on $\bP(\cE)$ and those on the base becomes weaker.
There we have the following statement, see \autoref{thm:bundles-surfaces} and \autoref{cor:P2} \& \ref{cor:hirzebruch}.

\begin{theoremalpha}
\label{main:surface}
Let $\pi \colon \bP(\cE) \to S$ be a projective bundle over a smooth projective surface,
and let $D = \pi^*A + H$ be very ample.
Then a locally free sheaf $\pi^* \cF(D)$ is Ulrich if and only if
\[
\sH^\bullet(S,\cF)=0 = \sH^\bullet(S, \cF \otimes \cO_S(-D')),
\]
where $D' = \rank(\cE) A + c_1(\cE)$.

In particular, there are Ulrich bundles of rank two with respect to $D$, if $S$ is the projective plane or a Hirzebruch surface.
\end{theoremalpha}

Note that if $D'$ is very ample on $S$, the locally free sheaf $\cF(D')$ from \autoref{main:surface} is an Ulrich bundle on $S$ with respect to $D'$.
So whenever there are Ulrich bundles known on the surface (for many polarisations),
then this implies the existence of an Ulrich bundle on $\bP(\cE)$.
By this we generalise results of \cite{Fania-etal} about the existence of Ulrich bundles on $\bP^1$-bundles over certain surfaces.
Note that the result here is not optimal:
for example, under specific conditions on the Chern classes of a locally free sheaf $\cE$ of rank two,
the corresponding $\bP^1$-bundles over $\bP^2$ admits Ulrich line bundles, see \cite[Prop. 5.1]{Fania-etal}.

We also obtain the existence of Ulrich bundles on blowups of projective space in a linear subspace, using that such a blowup carries the structure of a projective bundle, see \autoref{prop:ulrich-bundle-pn}.

Finally we note that the method presented here may serve to obtain easily more Ulrich bundles on projective bundles over surfaces different from the projective plane or a Hirzebruch surface, see also \autoref{rem:further-surfaces}.
But we also want to point out that the approach used here, does not generalise well for polarisations of the form $D = \pi^*A+nH$ with $n>1$.

\subsection*{Conventions}

Tensor products of a sheaf $\cF$ and a line bundle $\cO_X(D)$ are abbreviated as $\cF(D) \coloneqq \cF \otimes \cO_X(D)$.

For two sheaves $\cF$ and $\cG$ on a variety $X$, we  write $\Hom^\bullet(\cF,\cG)$ for the Hom complex
$\bigoplus_i \Ext^i_X(\cF,\cG)[-i]$,
which is a complex of vector spaces with zero differentials.
We use similar notation for the cohomology of a sheaf $\cF$, so $H^\bullet(X,\cF) = \Hom^\bullet(\cO_X,\cF)$.

Finally, all varieties here will be smooth and projective over an algebraically closed field of characteristic zero.

\subsection*{Acknowledgements}

The author wants to thank Klaus Altmann, Gianfranco Casnati, Francesco Malaspina, Peter Newstead, Joan Pons-Llopis and an anonymous referee for useful suggestions and comments.

\section{Preliminaries}

For the purposes of this work, we recall a cohomological characterisation of Ulrich bundles.

\begin{definition}
Let $X$ be a smooth projective variety
and $A$ a very ample divisor on $X$.
Then a locally free sheaf $\cF$ is called \emph{Ulrich bundle} with respect to $A$ if
\[
\sH^\bullet(X, \cF(-iA)) = 0
\]
for $i=1,\ldots,\dim(X)$.
\end{definition}

\begin{remark}
\label{rem:Serre-Ulrich}
If $\cF$ is an Ulrich bundle with respect to a very ample $A$ on a smooth projective variety $X$ of dimension $n$,
then also $\cF^\vee (K_X+(n+1)A)$ is Ulrich.
To see this note that by Serre duality
\[
\begin{split}
\sH^\bullet\bigl(X,\cF^\vee(K_X+(n+1-p)A\bigr) &=
\Hom^{n-\bullet}\bigl(\cF^\vee((n+1-p)A),\cO_X\bigr)^\vee = \\
& = \sH^{n-\bullet}\bigl(X,\cF((p-n-1)A)\bigr)^\vee
\end{split}
\]
and use that $\cF$ is Ulrich.
So whenever we construct an Ulrich bundle, we get automatically a second one.

Note that $\cF$ can coincide with $\cF^\vee (K_X+(n+1)A)$.
For example, if $\cF$ is of rank two, then $\cF^\vee \cong \cF(-c_1(\cF))$.
Hence for $\cF$ Ulrich with $c_1(\cF) = K_X + (n+1)A$, we find that
$\cF^\vee (K_X+(n+1)A) \cong \cF$.
If $X$ is a surface, then such Ulrich bundles are called \emph{special}; see also \cite{Eisenbud-Schreyer-Weyman, Casnati-q0}.
\end{remark}

\subsection{Projective bundles}

In the following, we collect some facts about projective bundles,
as can be found in \cite{Hartshorne}.

\begin{lemma}
\label{lem:bundles-canonical}
The canonical divisor of a projective bundle $\pi \colon \bP(\cE) \to X$ is given by
\[
K_{\bP(\cE)} = \pi^*(K_X + c_1(\cE)) - \rank(\cE)H,
\]
where $H$ is the relative hyperplane section.
\end{lemma}

Recall that $\pi$ is a flat morphism, so we do not run into problems pulling back divisors.

\begin{proof}
The relative canonical divisor for $\pi$ is given by
\[
K_{\bP(\cE)} \sim \pi^* K_X + K_{\bP(\cE)|X}.
\]
By \cite[Ex. III.8.4]{Hartshorne}, we have
\[
K_{\bP(\cE)|X} \sim \pi^* c_1(\cE) - \rank(\cE)H,
\]
from which follows the statement.
\end{proof}

\begin{lemma}[{\cite[Prop. II.7.10]{Hartshorne}}]
\label{lem:bundles-veryample}
Let $\pi \colon \bP(\cE) \to X$ be a projective bundle over a smooth projective variety $X$
with relative hyperplane section $H$.
If $A$ is an ample divisor on $X$, then there is an $n > 0$ such that $\pi^*(nA) + H$ is very ample.
\end{lemma}

Finally, we give the main cohomological ingredient to establish much of the cohomology vanishing.

\begin{proposition}[\cite{Orlov}]
\label{prop:orlov}
Let $\pi \colon \bP(\cE) \to X$ be a projective bundle over a smooth projective variety $X$.
Then there is a semi-orthogonal decomposition
\[
\Db(\bP(\cE)) = \sod{ \pi^* \Db(X) \otimes \cO_\bP(-nH), \ldots, \pi^* \Db(X) \otimes \cO_\bP(-H), \pi^* \Db(X)},
\]
where $n+1 = \rank(\cE)$ and $H$ is the relative hyperplane section.
\end{proposition}

\begin{remark}
For readers which are unfamiliar with derived categories, we want to point out the two consequences of the semi-orthogonal decomposition above.
\begin{enumerate}
\item
for coherent sheaves $\cF, \cG$ on $X$ we get
\[
\Hom^\bullet(\pi^*\cF, \pi^*\cG(-pH)) = 0
\]
for $p = 1, \ldots, n$. This is the semi-orthogonality of the decomposition, and is crucial for much of the cohomology vanishing that we will need.
\item
any coherent sheaf on $\bP(\cE)$ has an `approximation' by objects in the parts $\pi^* \Db(X) \otimes \cO_{\bP(\cE)}(-pH)$.
Put the other way, we can build sheaves on $\bP(\cE)$ using exact triangles whose terms are in these parts. This will be used in \autoref{lem:XXX}.
\end{enumerate}
For more information on derived categories, see the introductory text \cite{Hochenegger} or the excellent text book \cite{Huybrechts}.
\end{remark}

\begin{example}
\label{ex:hirzebruch}
Just to see this in the simplest case of a Hirzebruch surface $\bF_r = \bP(\cE)$ with $\cE = \cO_{\bP^1} \oplus \cO_{\bP^1}(r)$.
There we have two natural candidates for the relative hyperplane section, namely $C_\pm$ with the property that $C_\pm^2 = \pm r$.
Using a fibre $f$, we have that $C_+ \equiv rf + C_-$ and $\omega_{\bF_r} = \cO(-(r+2)f -2C_-) = \cO((r-2)f - 2C_+)$.
Here we find that $\pi^* K_{\bP^1} = -2f$ and $\pi^* \det(\cE) = \cO(rf)$.
Plugging all this into \autoref{lem:bundles-canonical}, we conclude that $H = C_+$.
Note that a divisor $D = af+bC_+$ on $\bF_r$ is very ample if and only if $a,b>0$.

Finally we remark that \autoref{prop:orlov} is indifferent about the choice between $C_-$ and $C_+$. For this note that $\cO_{\bF_r}( C_+ ) = \cO_{\bF_r}( C_- ) \otimes \pi^* \cO_{\bP^1}(r)$ and $\blank \otimes \pi^* \cO_{\bP^1}(r)$ is an autoequivalence of $\pi^*\Db(\bP^1)$.
\end{example}

\section{Projective bundles over arbitrary varieties}

In this section, we treat the case of Ulrich bundles on projective bundles over arbitrary varieties. The following statement will be central for the next sections, where we specialise to varieties of low dimension.

\begin{proposition}
\label{prop:bundles-general}
Let $\pi \colon \bP(\cE) \to X$ be a projective bundle over a smooth projective variety $X$,
where $\cE$ has rank $n+1$.
For a very ample divisor $D = \pi^* A + H$ with $H$ the relative hyperplane section and
a locally free sheaf $\cF$ on $X$ with $\sH^\bullet(X,\cF) = 0$,
the locally free sheaf $\pi^* \cF(D)$ is Ulrich if and only if
\[
\Hom^\bullet_{X}\bigl(\sym^k \cE, \cF(-c_1(\cE)-(n+1+k)A)\bigr) = 0
\]
for $k=0,\ldots,\dim(X)-2$.
\end{proposition}

This proposition generalises \cite[Thm. 2.1]{Fania-etal}, where essentially this characterisation is given for Ulrich line bundles on projective bundles.

\begin{proof}
We abbreviate $\bP \coloneqq \bP(\cE)$ in the following.
By \autoref{prop:orlov}, we have a semi-orthogonal decomposition
\[
\Db(\bP) = \sod{ \pi^* \Db(X) \otimes \cO_\bP(-nH), \ldots, \pi^* \Db(X) \otimes \cO_\bP(-H), \pi^* \Db(X)}.
\]
Let $\cF$ be a locally free sheaf on $X$ with $\sH^\bullet(X,\cF) = 0$.
So by the semi-orthogonal decomposition above we find that
\[
\sH^\bullet(\bP,\pi^* \cF (-iD)) = 0
\]
for $i=0,\ldots,n$.
Note that here $\dim(\bP) = n+m$, where $m = \dim(X)$.
So this is not sufficient to conclude that $\pi^*\cF(D)$ is an Ulrich bundle.
We need additionally the vanishing of
\[
\sH^\bullet\bigl(\bP,\pi^*\cF(-(n+1+k)D)\bigr).
\]
for $k=0,\ldots,m-2$.
By \autoref{lem:bundles-canonical} we get
\[
-(n+1)H = K_{\bP} - \pi^*(K_X + c_1(\cE))
\]
Using this for the above cohomology group we see that
\[
\begin{split}
&\sH^\bullet(\bP,\pi^* \cF(-(n+1+k)D)) =
\\ &=
\Hom^\bullet_{\bP}\bigl(\cO_{\bP}, \pi^*(\cF(-K_X-c_1(\cE)-(n+1+k)A)) \otimes \cO_{\bP}(K_{\bP}-kH)\bigr) =
\\ &=
\Hom^\bullet_{\bP}\bigl(\pi^*(\cF(-K_X-c_1(\cE)-(n+1+k)A)), \cO_{\bP}(kH)[n+m]\bigr)^\vee =
\\ &=
\Hom^\bullet_{\bP}\bigl(\cF(-K_X-c_1(\cE)-(n+1+k)A), \pi_*\cO_{\bP}(kH)[n+m]\bigr)^\vee =
\\ &=
\Hom^\bullet_{X}\bigl(\sym^k \cE, \cF(-c_1(\cE)-(n+1+k)A)[n]\bigr)
\end{split}
\]
where we use Serre duality two times, adjunction and $\pi_* \cO_\bP(kH) = \sym^k \cE$; see \cite[Prop. II.7.11]{Hartshorne}.
Note that these equalities are only valid on the level of derived categories, but here $\pi^*(\blank)$ and $\pi_* \cO_{\bP}(kH)$ coincide with their derived counterparts.
From this follows the claim.
\end{proof}

\begin{remark}
\label{rem:split-bundle}
For a locally free sheaf $\cE$ that splits into line bundles, the additional cohomology vanishing can be checked more easily in examples.
Let $\cE = \bigoplus_{i=0}^n \cO(D_i)$ and $\pi \colon \bP(\cE) \to X$,
and moreover, let $D = \pi^*A + H$ be very ample.
By \autoref{prop:bundles-general}, the additional conditions on a locally free sheaf $\cF$ with $\sH^\bullet(X,\cF)=0$ to be Ulrich amount to the vanishing of
\[
\sH^\bullet\Biggl(X, \cF\Bigl(-\sum_{i=0}^n D_i - \sum_{j=1}^k D_{i_j} -(n+1+k)A\Bigr) \Biggr)
\]
for all $0 \leq i_1 < \cdots < i_k \leq n$ and $k = 0,\ldots,\dim(X)-2$.

A result based on this remark is \autoref{prop:ulrich-bundle-pn}.
\end{remark}

\begin{remark}
Let $\pi \colon \bP(\cE) \to X$ be a projective bundle and $D = \pi^* A + H$ very ample.
For $\cG$ an Ulrich bundle with respect to $D$, one might ask whether there is a locally free sheaf $\cF$ on $X$ with $\sH^\bullet(X,\cF) = 0$, such that $\cG = \pi^* \cF(D)$.

In the following, we use knowledge about Ulrich line bundles on Hirzebruch surfaces, see \autoref{ex:hirzebruch-ulrich} for details.
First note that for these Ulrich line bundles, the answer to the above question is positive.
Now, consider the polarisation $D = f + C_+ = 2f+C_-$ for $\bF_1$.
There is a non-trivial extension of two Ulrich line bundles:
\[
0 \to \cO_{\bF_1}(C_+) \to \cG \to \cO_{\bF_1}(2f) \to 0.
\]
By \cite[Prop. 5.4]{Casanellas-Hartshorne},
$\cG$ is an indecomposable Ulrich bundle of rank two, and there is no other such locally free sheaf on $\bF_1$.
But $\cG$ cannot be of the form $\pi^* \cF(D)$, as all locally free sheaves on $\bP^1$ split into line bundles.
So the answer to the question above is negative in general.

Note that $\cG$ is an extension of the Ulrich line bundles $\cO_{\bF_1}(C_+)$ and $\cO_{\bF_1}(2f)$ for the same polarisation.
So we can ask whether all Ulrich bundles $\cG$ with respect to a polarisation $\pi^*A + H$ arise as extensions of Ulrich bundles of the shape $\pi^*\cF(D)$. But even this is not the case in general.

To see this, we consider a rational normal scroll $\bP(\cE) \to \bP^1$ where $\cE = \bigoplus_{i=0}^n \cO_{\bP^1}(a_i)$ and $a_i > 0$.
By \cite[Lem. 3.2]{Aprodu-HML}, the locally free sheaves $\Omega^j_{\bP(\cE)|\bP^1}(j,j+1)$ are stable Ulrich bundles with respect to $H$, for $j=0,\ldots,n$.
Hence, for $j>0$, these Ulrich bundles are neither obtained by pullbacks twisted by a divisor nor extensions thereof.
\end{remark}

\section{Projective bundles over curves}
\label{sec:curves}

In this section, we specialise to the question about Ulrich bundles on projective bundles over curves.

The following lemma is well-known. For the convenience of the reader, we give a proof.

\begin{lemma}
\label{lem:curves-immaculate}
Let $C$ be a projective curve of genus $g$.
Then $\sH^\bullet(C,\cL) = 0$ for a general line bundle $\cL$ of degree $g-1$.
\end{lemma}

\begin{proof}
Note that $\Pic^{g-1}(C)$ has dimension $g$, in particular, a general line bundle $\cL$ of degree $g-1$ is not effective, so $\sH^0(C,\cL)=0$.
The degree of $\cL$ was chosen in such a way that $\sH^1(C,\cL)=0$
by the Riemann--Roch theorem.
\end{proof}

\begin{remark}
Note that \autoref{lem:curves-immaculate} gives rise to many Ulrich line bundles on curves.
More generally, let $A$ be a very ample divisor and $\cE$ a locally free sheaf on $C$.
Then $\cE(A)$ is Ulrich with respect to $A$ if and only if $\sH^\bullet(C,\cE)=0$.

Hence the study of Ulrich bundles can be reduced to the search of locally free sheaves with no cohomology at all.
Except for $C = \bP^1$, one can construct many indecomposable Ulrich bundles of higher rank;
for more details on the following classification, see \cite[\S 4.1]{Coskun-Intro}.

For $g=0$, that is, $C = \bP^1$, the only line bundle with no cohomology at all is $\cO_{\bP^1}(-1)$.
Recall that any locally free sheaf $\cE$ on $\bP^1$ splits into line bundles. So if $\cE$ is such that $\sH^\bullet(\bP^1,\cE)=0$, then it is isomorphic to $\bigoplus \cO_{\bP^1}(-1)$.

For $g=1$, we recall \cite[Thm. 5]{Atiyah}: for each $r \geq 1$, there is an indecomposable locally free sheaf $\cF_r$ of rank $r$ and of degree zero with $\sH^\bullet(C,\cF_r) \cong \kk$, additionally $\cF_r$ is unique up to isomorphism. Moreover, for $\cF_r \otimes \cL$ with $\cL$ a non-trivial line bundle of degree zero, we have $\sH^\bullet(C,\cF_r \otimes \cL) = 0$.
Hence, any locally free sheaf $\cE$ on an elliptic curve with $\sH^\bullet(C,\cE)=0$ splits into a direct sum of such $\cF_r \otimes \cL$.

For $g>2$, one can consider a direct sum of line bundles $\bigoplus \cL_i$ with $\sH^\bullet(C,\cL_i)=0$. Then a general deformation of this direct sum will yield an indecomposable locally free sheaf $\cE$ with $\sH^\bullet(C,\cE)=0$.
\end{remark}

As an application of \autoref{prop:bundles-general}, we get the following statement.

\begin{theorem}
\label{thm:bundles-curves}
Let $\pi \colon \bP(\cE) \to C$ be a projective bundle over a smooth projective curve $C$.
Moreover, let $D = \pi^* A + H$ be a very ample divisor with $H$ the relative hyperplane section.

A locally free sheaf $\pi^* \cF(D)$ is Ulrich on $\bP(\cE)$ with respect to $D$ if and only if $\cF$ is a locally free sheaf on $C$ with
$\sH^\bullet(C,\cF) = 0$.
In particular, for any very ample $A'$, $\cF(A')$ is Ulrich with respect to $A'$.
\end{theorem}

\begin{remark}
\label{rem:bundles-curves}
By \autoref{lem:bundles-veryample}, very ample divisors of the shape $\pi^* A + H$ exist on $\bP(\cE)$.
Moreover, by \autoref{lem:curves-immaculate}, there are line bundles $\cL$ on $C$ such $\sH^\bullet(C,\cL) = 0$.
In particular, a projective bundle over a curve admits an abundance of Ulrich line bundles.
\end{remark}

\begin{remark}
The above result generalises \cite[Thm. 2.1]{Aprodu-etal}, where Ulrich line bundles on geometrically ruled surfaces over curves are classified.
Note that in \cite{Aprodu-etal}, also Ulrich bundles of rank two are constructed for any given polarisation of the ruled surface. This uses the Serre construction of locally free bundles, which does not generalise that easily to higher dimensions.

In \cite[Thm. 4.7, Cor. 4.10]{Aprodu-HML}, Ulrich bundles on rational normal scrolls $\bP\bigl( \bigoplus_i \cO_{\bP^1}(a_i) \bigr)$ with respect to polarisations of the form $\pi^*A +H$ are characterised. There it is shown that there are at most finitely many.

Moreover, in \cite[Thm. B]{Faenzi-Malaspina}, rigid Ulrich bundles on Hirzebruch surfaces of degree $\geq 5$ are described as extensions (more precisely: mutations) of Ulrich line bundles. This gives a curious connection between such Ulrich bundles and Fibonacci numbers.
Additionally, in the case that such a Hirzebruch surface is of degree $4$,
aCM sheaves of even rank are classified: there is a $\bP^1$-family of such sheaves, see \cite[Thm. A]{Faenzi-Malaspina}.

Finally, in \cite{Antonelli-Hirzebruch} a full classification of Ulrich bundles on Hirzebruch surfaces for any given polarisation is obtained.
\end{remark}

The following example can also be obtained as a consequence of the more general \cite[Thm. 2.1]{Aprodu-etal}.

\begin{example}
\label{ex:hirzebruch-ulrich}
Let
$\bF_r = \bP(\cO_{\bP^1} \oplus \cO_{\bP^1}(r))$ be the $r$-th Hirzebruch surface with $r \geq 0$ and let $\pi \colon \bF_r \to \bP^1$ be the projection.
Up to isomorphism, the only line bundle $\cL$ on the base $\bP^1$ with $\sH^\bullet(\bP^1,\cL) = 0$ is $\cO_{\bP^1}(-1)$.

If we denote by $f$ a fibre of $\pi$ and by $C_+$ the rational curve on $\bF_r$ with $C_+^2 = r$,
then for $\cO_{\bF_r}(f) = \pi^* \cO_{\bP^1}(-1)$ we find that $\sH^\bullet(\bF_r, \cO_{\bF_r}(f)) = 0$.
Additionally, one can check that $D = af + C_+$ is ample, which for smooth projective toric varieties is equivalent to being very ample.
Hence \autoref{thm:bundles-curves} and \autoref{rem:Serre-Ulrich} imply that for this polarisation, the line bundles
\[
\cO_{\bF_r}((a-1)f+C_+)
\quad\text{and}\quad
\cO_{\bF_r}((r-1+2a)f)
\]
are Ulrich.

For $r>0$ these give all Ulrich line bundles. To see this note that
\[
\cO_{\bF_r}(-f),\
\cO_{\bF_r}(if - C_+) \text{ with $i \in \bZ$},\
\cO_{\bF_r}((r-2)f-2C_+),
\]
is a complete list of line bundles $\cL$ on $\bF_r$ with the property that $\sH^\bullet(\bF_r,\cL)=0$.
Therefore, the very ample divisor $D = af + bC_+$ with $a>0$ and $b>1$ does not give rise to any Ulrich line bundle.

For $r=0$, we find that $\bF_0 = \bP^1 \times \bP^1$ has precisely two families of line bundles $\cL$ with $\sH^\bullet(\bP^1\times\bP^1,\cL) = 0$:
\[
\cO_{\bF_0}(if-f'),\
\cO_{\bF_0}(-f+if') \text{ with $i \in \bZ$}
\]
where $f' = C_+$ is a fibre of the second ruling.
Hence given any very ample divisor $D = af+bf'$ with $a,b>0$, we get that
\[
\cO_{\bF_0}((a-1)f+(2b-1)f')
\quad\text{and}\quad
\cO_{\bF_0}((2a-1)f+(b-1)f')
\]
are the Ulrich line bundles with respect to $D$.
\end{example}

\section{Projective bundles over surfaces}

In this section, we specialise to the question about Ulrich bundles on projective bundles over surfaces. Later we pay more attention to the case of the projective plane and Hirzebruch surfaces.

\begin{theorem}
\label{thm:bundles-surfaces}
Let $\pi \colon \bP(\cE) \to S$ be a projective bundle over a smooth projective surface $S$.
Moreover, let $D = \pi^* A + H$ be a very ample divisor with $H$ the relative hyperplane section.

A locally free sheaf $\pi^* \cF(D)$ is Ulrich on $\bP(\cE)$ with respect to $D$ if and only if $\cF$ is a locally free sheaf on $S$ with
\[
\sH^\bullet(S,\cF) = 0 = \sH^\bullet(S,\cF(-D'))
\]
where $D' \coloneqq \rank(\cE)\cdot A + c_1(\cE)$.
In particular, if $D'$ is very ample, then $\cF(D')$ is Ulrich with respect to $D'$.
\end{theorem}

This statement generalises \cite[Thm. 2.4]{Fania-etal}, where the case of $\bP^1$-bundles over surfaces is treated.
Recall that by \autoref{lem:bundles-veryample}, a very ample divisor of the form $D = \pi^*A + H$ does always exist.

\begin{proof}
This is a direct application of \autoref{prop:bundles-general}.

\end{proof}

\begin{question}
\label{q:very-ampleness-bundle-base}
Already the connection between the ample divisors on a surface and ample divisors on projective bundles over the surface is quite complicated, see \cite{Misra-Ray}.
So it is not obvious, whether $D = \pi^*A + H$ (very) ample implies that also $D' = (n+1)A + c_1(\cE)$ is (very) ample, as well.
Examples suggest that this implication is true, but also show that the other implication is wrong in general; see \autoref{ex:blowupP3}.

On the other hand, by \autoref{lem:bundles-veryample}, we can choose $A$ positive enough such that $D$ and $D'$ are very ample.
\end{question}

\begin{example}
In \cite{Fania-etal}, many more Ulrich bundles of low rank are constructed on $\bP^1$-bundles over several surfaces, obtaining even a full classification in those cases of Ulrich line bundles.
\end{example}

\subsection{Projective bundles over $\bP^2$}

\begin{proposition}[{\cite{Coskun-Genc, Costa-Miro-Roig}}]
\label{prop:ulrich-p2}
For $d>0$, there are are Ulrich bundles on $\bP^2$ with respect to $\cO_{\bP^2}(d)$:
\begin{itemize}
\item[$d=1$:] direct sums of $\cO_{\bP^2}$;
\item[$d=2$:] direct sums of $\cT_{\bP^2}$;
\item[$d>2$:] locally free sheaves of rank at least $2$. For example, an Ulrich bundle $\cE$ of rank two fits into the short exact sequence
\[
0 \to \cO_{\bP^2}^{\oplus (d-1)}(d-2) \to \cO_{\bP^2}^{\oplus (d+1)}(d-1) \to \cE \to 0,
\]
which in the case of $d=2$ yields $\cT_{\bP^2}$.
\end{itemize}
\end{proposition}

The following lemma will be the basis for the construction of Ulrich bundles.

\begin{lemma}
\label{lem:XXX}
Let $d$ be a non-negative integers.
On $\bP^n$, there are locally free sheaves $\cF$ of rank $n$ or ${n+d \choose n-1}$ with
the properties that
\[
\sH^\bullet(\bP^n,\cF) = 0
\quad \text{and} \quad
\cF \in \sod{ \cO_{\bP^n}(d), \cO_{\bP^n}(d+1)}.
\]
\end{lemma}

\begin{proof}
The easiest way to guarantee the second condition of the statement is to assume that $\cF$ fits into a short exact sequence
\[
0 \to \cF \to \cO(d)^{\oplus b_1} \xto{\alpha} \cO(d+1)^{\oplus b_2} \to 0,
\]
as for a surjective morphism between locally free sheaves, the kernel is automatically locally free.
By applying $\sH^\bullet(\bP^n,\blank)$ to this sequence, we obtain
\[
\begin{split}
0 &\to \sH^0(\bP^n,\cF) \to \sH^0(\bP^n,\cO(d))^{\oplus b_1} \xto{\sH^0(\alpha)} \sH^0(\bP^n,\cO(d+1))^{\oplus b_2} \to \\
&\to \sH^1(\bP^n,\cF) \to 0.
\end{split}
\]
Here we want to ensure that $\sH^0(\alpha)$ is an isomorphism to conclude that $\sH^\bullet(\bP^n,\cF) = 0$.
A necessary condition for this to happen is that
$b_1 \cdot \dim \sH^0(\bP^n,\cO(d)) = b_2 \cdot \dim \sH^0(\bP^n,\cO(d+1))$, or more explicitly that
\[
b_1 \cdot {n+d \choose n} = b_2 \cdot {n+d+1 \choose n}
\quad \iff \quad
(d+1) b_1 = (n+d+1) b_2
\]

In particular, $b_1 = n+d+1$ and $b_2 = d+1$ is a valid solution.
There are solutions with smaller values, if $\gcd(n,d+1) > 1$,
but for this choice we find that there a generic choice of
$\alpha \colon \cO(d)^{n+d+1} \to \cO(d+1)^{d+1}$
induces an isomorphism $\sH^0(\alpha)$. For example, we can take
\[
\cO^{n+d+1}(d)
\xto{
\left(
\def\dots[#1]{ \ar[#1, dotted, no head, shorten=1mm]}
\begin{tikzcd}[row sep=tiny, column sep=small, ampersand replacement=\&]
x_0 \dots[rrrddd] \dots[rr] \&\& x_n \dots[rrrddd] \& 0 \dots[rr] \dots[rrdd] \&\& 0 \dots[dd] \\
0 \dots[rrdd] \dots[dd]\\
 \& \&\& \&\& 0 \\
0 \dots[rr] \&\& 0 \& x_0 \dots[rr] \&\& x_n
\end{tikzcd}
\right)
}
\cO^{d+1}(d+1)
\]
which is a direct generalisation of \cite[Prop.\ 5.9]{Eisenbud-Schreyer-Weyman}.
Hence for $b_1 = n+d+1$ and $b_2 = d+1$, we obtain a locally free sheaf $\cF$ of rank $n$ with
\[
\sH^\bullet(\bP^n,\cF) = 0, \quad
\cF \in \sod{ \cO_{\bP^n}(d), \cO_{\bP^n}(d+1)}.
\]

Another choice for a solution is
$b_1 = {n+d+1 \choose n}$ and $b_2 = {n+d \choose n}$.
By the same kind of argument, we obtain for this choice,
a locally free sheaf $\cF$ of rank $b_1-b_2 = {n+d \choose n-1}$.
Here, we can construct also an explicit locally free sheaf.
For this, consider the Euler sequence
\[
0 \to \Omega \to \cO(-1)^{n+1} \to \cO \to 0.
\]
Taking the $(d+1)$-th symmetric power of this short exact sequence and tensoring with $\cO(2d+1)$ yields
\[
0 \to \sym^{d+1} \Omega(2d+1) \to \cO(d)^{\oplus {n+d+1 \choose n}} \to \cO(d+1)^{\oplus {n+d \choose n}} \to 0.
\]
By construction $\sym^{d+1} \Omega(2d+1) \in \sod {\cO(d),\cO(d+1)}$, and taking cohomology of the last short exact sequence shows also that $\sH^\bullet(\bP^n, \sym^{d+1} \Omega(2d+1))=0$.
\end{proof}

\begin{remark}
\label{ex:ulrich-p2}
Let $\cF$ be a locally free sheaf on $\bP^2$.
The conditions of \autoref{lem:XXX} amount to $\cF(d+2)$ being an Ulrich bundle with respect to $\cO(d+2)$. To see this, note that $0 = \sH^\bullet(\bP^2,\cF(-d-2)) = \Hom^\bullet(\cO(d+2),\cF)$ if and only if $\cF \in \cO(d+2)^\perp = \sod{\cO(d),\cO(d+1)}$.

On $\bP^2$, note that in the case $d=0$, we obtain the Ulrich bundle
\[
\cF(2) \cong \Omega_{\bP^2}(3) \cong \cT_{\bP^2}
\]
of rank $2$ on $\bP^2$.
Actually, this is the only indecomposable Ulrich bundle with respect to $\cO_{\bP^2}(2)$, see \cite[Ex. 3.1]{Aprodu-HML}.
Moreover, we get for any $d > 0$ many Ulrich bundles of rank two. The construction in the proof of \autoref{lem:XXX} is the dual of the one given in \cite[Prop.\ 5.9]{Eisenbud-Schreyer-Weyman}.

Also note that the argument of \autoref{lem:XXX} can be extended to cover also the case $d=-1$, where we end up with the the Ulrich line bundle $\cO_{\bP^2}$.
\end{remark}

\begin{remark}
On $\bP^2$, in the case $d=1$, the proof of \autoref{lem:XXX} gives the locally free sheaf $\cF = \sym^2 \Omega_{\bP^2}(3)$, which appears in \cite[\S 3.4]{Hochenegger-Meachan} in a completely different context.
There the interest lies on ${}^\perp \cO_{\bP^2} \cap \cO_{\bP^2}^\perp$, which is a non-admissible subcategory of $\Db(\bP^2)$.
Note that the locally free sheaves inside ${}^\perp \cO_{\bP^2} \cap \cO_{\bP^2}^\perp$ are precisely the Ulrich bundles on $\bP^2$ with respect to the anticanonical polarisation $\cO_{\bP^2}(3)$.
\end{remark}

The following statement is a consequence of \autoref{thm:bundles-surfaces} and \autoref{lem:XXX}, by taking into account that for $D = \pi^* A + H$ we can replace $A$ by a more positive divisor, in order to guarantee that $D' = \rank(\cE)\cdot A + c_1(\cE)$ is very ample, see \autoref{lem:bundles-veryample}.

\begin{corollary}
\label{cor:P2}
Let $\pi \colon \bP(\cE) \to \bP^2$ be a projective bundle and let $D = \pi^*A+H$ be a very ample divisor such that $D' = \rank(\cE)\cdot A + c_1(\cE)$ is very ample, as well.
Then there are Ulrich bundles of rank two on $\bP(\cE)$ with respect to $D$.
\end{corollary}

\begin{remark}
This corollary generalises the results from \cite[\S 5]{Fania-etal} about Ulrich bundles of rank two on $\bP^1$-bundles over $\bP^2$.
Note that in \cite[Prop. 5.1]{Fania-etal}, it is shown under which restrictions on the Chern classes of $\cE$, there are Ulrich line bundles.
\end{remark}

\begin{question}
This is again \autoref{q:very-ampleness-bundle-base}:
If $D = \pi^*A +H$ is very ample, do we have at least in the situation of the theorem above, that $D' = (n+1)A+c_1(\cE)$ is very ample?
What is the minimal degree $d_{\min}$ of $D'$ in dependence of the possible choices for $A$?
In the case that $d_{\min}$ is always positive, we have a connection between Ulrich bundles on $\bP(\cE)$ and those on $\bP^2$.
In situations where $d_{\min}=1$ is possible, $\pi^*\cO_{\bP^2}(-1) \otimes \cO_{\bP(\cE)}(D)$ would be an Ulrich line bundle on $\bP(\cE)$.
\end{question}

\begin{example}
\label{ex:blowupP3}
Let $X = (\bP^3)'$ be the blowup of $\bP^3$ in a point.
There is a map $\pi \colon X \to \bP^2$, realising $X$ as a projective bundle.
More precisely, one has $X = \bP(\cE)$ with $\cE = \cO_{\bP^2}(1) \oplus \cO_{\bP^2}$.
Denoting by $h$ the pullback of a hyperplane section of $\bP^2$ via $\pi$, a divisor $D = ah+H$ is very ample if and only if $a > 0$.

By \autoref{thm:bundles-surfaces}, a locally free sheaf $\pi^*\cF(ah+H)$ is Ulrich with respect to $\cO_X(ah+H)$ if and only if $\cF(2a+1)$ is Ulrich on $\bP^2$ with respect to $\cO_{\bP^2}(2a+1)$.
For the minimal choice $a=1$, we therefore have Ulrich bundles of rank at least two, see \autoref{ex:ulrich-p2}.

Note that for the excluded value $a=0$, $\cO_{\bP^2}$ is an Ulrich line bundle on $\bP^2$ with respect to the very ample $\cO_{\bP^2}(1)$.
Moreover, for $\cF = \cO_{\bP^2}(-1)$, the cohomolgy groups $H^\bullet(X,\cL(-pD))$ vanish for the line bundle $\cL \coloneqq \pi^*\cF(H) = \cO_X(H-h)$ and $p=1,2,3$.
But $\cO_X(H)$ is not even ample, so $\cO_X(H-h)$ is not Ulrich.

For a deeper picture about Ulrich bundles on $(\bP^3)'$, see \cite{Casnati-etal}.
\end{example}

\begin{example}
\label{ex:blowupP3b}
Consider $\cE = \cO_{\bP^2}(1) \oplus \cO_{\bP^2}^{\oplus d}$ and
the projective bundle $\pi \colon \bP(\cE) \to \bP^2$.
As before, by \autoref{thm:bundles-surfaces},
a locally free sheaf $\pi^*F(ah + H)$ is Ulrich with respect to $ah + H$ if and only if $\cF((d + 1)a + 1)$ is Ulrich on $\bP^2$ with respect to $\cO_{\bP^2}((d + 1)a + 1)$.
Again, the case $a = 0$ is not possible, as $H$ is not even ample,
so we conclude that there are Ulrich bundles on X which have rank at least two.
\end{example}

\begin{proposition}
\label{prop:ulrich-bundle-pn}
Let $\cE = \cO_{\bP^n}(1) \oplus \cO_{\bP^n}^{\oplus d}$ and consider the projective bundle $\pi \colon \bP(\cE) \to \bP^n$.
Then there are Ulrich bundles of rank $n$ on $\bP(\cE)$ with respect to
the very ample divisor $D = h+H$, where $h$ is the pullback of the hyperplane section of $\bP^n$.
\end{proposition}

\begin{proof}
We show that there is a locally free sheaf $\cF$ of rank $n$ on $\bP^n$,
such that $\pi^*\cF(D)$ is an Ulrich bundle on $\bP(\cE)$ with respect to $D$.
For this we will use \autoref{prop:bundles-general}
in its formulation for split locally free sheaves in \autoref{rem:split-bundle}.
The conditions for $\cF$ to fulfill are that $\sH^\bullet(\bP^n,\cF)=0$ and
$\sH^\bullet(\bP^n,\cF(-1-\delta-(n+1+k))) = 0$
with $\delta \in \{0,1\}$ and $0 \leq k \leq n-2$.
The latter vanishing can be expressed as
\[
\Hom^\bullet(\cO(n+2+k),\cF) = 0
\]
for $0 \leq k \leq n-1$.
Using that $\Db(\bP^n)=\sod{\cO(n),\cO(n+1),\cO(n+2),\ldots,\cO(2n+1)}$,
these conditions become
\[
\sH^\bullet(\bP^n,\cF) = 0
\quad\text{and}\quad
\cF \in \sod{\cO(n),\cO(n+1)}.
\]
So by \autoref{lem:XXX} there are locally free sheaves $\cF$ of rank $n$ on $\bP^n$ that satisfy these conditions.
In turn, this gives Ulrich bundles $\pi^*F(D)$ of rank $n$ on $\bP(\cE)$ with respect to $D=h+H$ by \autoref{prop:bundles-general}.
\end{proof}

\begin{remark}
For $\cE = \cO_{\bP^n}(1) \oplus \cO_{\bP^n}^{\oplus d}$, the projective bundle $\bP(\cE)$ is isomorphic to the blowup $(\bP^{n+d})'$ of $\bP^{n+d}$ in a $(d-1)$-dimensional linear subspace.
So in \autoref{ex:blowupP3} \& \ref{ex:blowupP3b} and \autoref{prop:ulrich-bundle-pn},
there is also the possibility to construct Ulrich bundles via pullback using the map $(\bP^{n+d})' \to \bP^{n+d}$,
by using \cite[Thm. 3.1]{Casnati-Kim}.
\end{remark}

\subsection{Projective bundles over Hirzebruch surfaces}

\begin{proposition}[\cite{Aprodu-etal}]
Let $\pi \colon \bF_r \to \bP^1$ be the projection of Hirzebruch surface $\bF_r = \bP(\cO \oplus \cO(r))$ to $\bP^1$.
For any very ample $D = ah + bH$, where $h$ is the pullback of the hyperplane section of $\bP^1$, there exist
\begin{itemize}
\item Ulrich line bundles if and only if $b=1$,
\item Ulrich bundles of rank two if $b>1$
\end{itemize}
on $\bF_r$ with respect to $D$.
\end{proposition}

\begin{remark}
In \cite{Antonelli-Hirzebruch}, Ulrich bundles of arbitrary rank are described as cokernels of an injective map of split locally free sheaves.
\end{remark}

Again as an application of \autoref{thm:bundles-surfaces}, we arrive at the following statement.

\begin{corollary}
\label{cor:hirzebruch}
Let $\pi \colon \bP(\cE) \to \bF_r$ be a projective bundle and let $D = \pi^*A+H$ be a very ample divisor such that $D' = \rank(\cE)\cdot A + c_1(\cE)$ is very ample, as well.
Then there are Ulrich bundles of rank two on $\bP(\cE)$ with respect to $D$.
\end{corollary}

\begin{remark}
For the base variety $\bF_0 = \bP^1 \times \bP^1$, there are Ulrich line bundles for any polarisation, so in this case we have even Ulrich line bundles on $\bP(\cE)$.

For base variety $\bF_r$ with $r>0$, the question remains under which conditions on $\cE$ there are Ulrich line bundles on $\bP(\cE)$.
Note that in \cite[Thm. 5.9]{Fania-etal}, it is shown that there are Ulrich bundles of rank two on a $\bP^1$-bundle over $\bF_1$.
\end{remark}

\begin{remark}
\label{rem:further-surfaces}
The method described in this section, establishes the existence of Ulrich bundles on projective bundles over a surface, where sufficiently many Ulrich bundles are known.
We give a list of such surfaces, which is far from exhaustive, and some references:
\begin{itemize}
\item del Pezzo surfaces: \cite{Pons-Tonini, Coskun-Pezzo};
\item K3 surfaces: \cite{Coskun-K3, Watanabe, Aprodu-Farkas-Ortega, Casnati-Galuzzi};
\item surfaces with $p_q = 0$ and $q=0,1$: \cite{Casnati-q0, Casnati-q1}.
\end{itemize}
Moreover, most of the results of \cite{Fania-etal} about Ulrich bundles on $\bP^1$-bundles over certain surfaces should allow a generalisation to arbitrary projective bundles over these surfaces.
We leave the search for further Ulrich bundles to the interested readers.
\end{remark}

\input{biblio}

\BgThispage

\end{document}

config: arxiv


TITLE: Find the value of $3x^4+14x^3+24x^2-6x-10$, when $x=-2+\sqrt{3}i$?
QUESTION [0 upvotes]: What will be the value  of $f(x)=3x^4+14x^3+24x^2-6x-10$, when $x=-2+\sqrt{3}i$?
I can evaluate multiplying 4 times. I am looking for a shortcut method?

REPLY [3 votes]: Hint $\ x^2+4x+7 = 0 $ so by division
$$\smash[t]{f(x) =  (3x^2+2x-5)\overbrace{(x^2+4x+7)}^{\large 0} + 25 = 25}$$
Remark $ $ If your know modular arithemtic then it is simpler to compute $f(x)\bmod x^2+4x+7,\,$ i.e. use $\,x^2 \equiv -4x-7\,$ to reduce all $x^n,\, n> 2$
This is a generalization of the Remainder Theorem  $\ f(a) = f(x)\bmod (x-a)$


TITLE: Proving that union of two equivalence relations may not be equivalence but intersection is always
QUESTION [1 upvotes]: How do we show that if $R$ and $S$ are two equivalence relations, then $R\cup S$ may/may not be a equivalence relation but $R\cap S$ is always a equivalence relation rigorously.

i can show it by example as take $A$ set to be $\{1,2,3\}$, then $R=\{ (1,1),(2,2),(3,3),(1,2),(2,1)\}$ and $S = \{(1,1),(2,2),(3,3),(1,3),(3,1)\}$ are both equivalence but $R\cup S$ is not since it needs to have $(2,3)$ and $(3,2)$ in it too but intersection contains the identity one which is equivalence relation.

REPLY [1 votes]: Rigorous proof
Is $R\cup S$ and/or $R\cap S$ reflexive?
If $a\in A$ then $(a,a)\in R$ and $(a,a) \in S$ because both are equiv.  So $(a,a) \in R\cup S$ and $(a,a)\in R\cap S$.  So both are reflexive.
Is $R\cup S$ and/or $R\cap S$ symmetric?
Suppose $(a,b) \in R\cup S$.  Then either $(a,b) \in R$ or $(a,b) \in S$ (or both).  If the first case then $(b,a)\in R$ as $R$ is reflexive. And in the second case $(b,a) \in S$ as $S$ is reflexive.  So either $(b,a) \in R$ or $(b,a) \in S$.  So $(b,a) \in R\cup S$ and $R\cup S$ is symmetric.
If $(a,b) \in R\cap S$ then $(a,b) \in R$ and $(a,b) \in S$.  As both are equivalent rel, we must have $(b,a) \in R$ and we must have $(b,a) \in S$ so $(b,a) \in R \cap S$.  So $R\cap S$ is symmetric.
It's transitive where things get interesting.  Do you have an intuition that if $(a,b),(b,c)$ are both in $R\cap S$ then $(a,c) \in R\cap S$?  But do you have intuition why that will go pear shape if $(a,b),(b,c)\in R\cup S$ that $(a,c)$ need not be?  (Hint: what if $(a,b)\in R$ but not $S$ and $(b,c)\in S$ but not $R$?)


TITLE: Scale factor evolution for positive curvature universe
QUESTION [1 upvotes]: From the Friedmann equations we can write
$$\dot{a}^2=H_0^2\left(\Omega_{R,0}a^{-2}+\Omega_{M,0}a^{-1}+(1-\Omega_0)+\Omega_{\Lambda,0}a^2\right).$$
Typically, one takes the square root of both sides and for $\Omega_0\leq 1$ this differential equation can be solved to obtain $a(t)$ (sometimes of course requiring numerics).
For $\Omega_0>1$, however, this leads us to a solution where $a(t)$ is now complex. I think I've missed something crucial but I can't seem to find it. How do I make sense of the complex solution?

REPLY [1 votes]: Let's discuss your example $\Omega_{R,0}=\Omega_{\Lambda,0}=0$, viz.$$\int H_0dt=\pm\int\frac{ada}{\sqrt{\frac{\Omega_{M,0}^2}{4(\Omega_0-1)}-(\Omega_0-1)\left(a-\frac{\Omega_{M,0}}{2(\Omega_0-1)}\right)^2}}.$$If $\Omega_{M,0}>0,\,\Omega_0>1$, the substitution$$a=\frac{\Omega_{M,0}}{2(\Omega_0-1)}(1+\sin\theta)$$obtains$$\int H_0dt=\pm\int\frac{\Omega_{M,0}}{2(\Omega_0-1)^{3/2}}\left(1+\sin\theta\right)\operatorname{sgn}(\cos\theta)d\theta.$$We must enforce $\theta\in\Bbb R$, so in a positive-curvature universe $a$ is capped.


TITLE: How to prove $2^n\over n^2$ diverges?
QUESTION [2 upvotes]: Hi i was solving an exercises and now im stuck at the part where i have to show
that this diverges. It is kind of obvious but i still would want to prove it.

REPLY [0 votes]: $$ \lim_{n\rightarrow\infty}\frac{2^n}{n^2} =\frac{\infty}{\infty}\,{\small\{\text{L'Hopital's}\}} =\frac{\log{2}}{2}\,\lim_{n\rightarrow\infty}\frac{2^n}{n} =\frac{\infty}{\infty}\,{\small\{\text{L'Hopital's}\}} =\frac{\log^2{2}}{2\cdot1}\,\lim_{n\rightarrow\infty}2^n \color{red}{\rightarrow\infty} $$

Generally:
$$ \lim_{n\rightarrow\infty}\frac{a^n}{n^k} =\frac{\log{a}}{k}\,\lim_{n\rightarrow\infty}\frac{a^n}{n^{k-1}} =\frac{\log^2{a}}{k(k-1)}\,\lim_{n\rightarrow\infty}\frac{a^n}{n^{k-2}} =\,\cdots\, =\frac{\log^k{a}}{k!}\,\lim_{n\rightarrow\infty}a^n \color{red}{\rightarrow\infty} $$


text:
\section{Conclusion}
\label{sec: conclusion}
In this paper, we presented a computationally tractable approach for constrained MHE of discrete-time linear systems. An anytime pMHE iteration scheme is proposed in which a state estimate at each time instant is computed based on an arbitrary number of optimization algorithm iterations. The underlying optimization algorithm  consists of a mirror descent-like method which generalizes the gradient descent and can therefore be executed quickly. Under suitable assumptions on the Bregman distance and the step sizes, GES of the estimation errors was  established and is ensured after any  number of optimization
algorithm iterations. In addition, the performance of the iteration scheme was characterized by the resulting real-time regret for which upper bounds were derived. The proposed iteration scheme provides stable estimates after each optimization iteration and possesses a sublinear regret which can be rendered
arbitrarily small by increasing the number of iterations.\\
The proposed anytime pMHE iteration scheme is conceptually related to the anytime model predictive control (MPC) iteration scheme with relaxed barrier
functions~\cite{feller2017stabilizing,feller2018sparsity}, where stabilizing control inputs are generated  after any number of optimization iterations. Our goal in future research is to combine both the MPC and MHE iteration schemes in an overall anytime
estimation-based MPC algorithm. Furthermore, comparisons to  real-time MHE techniques established in the literature and a further exploration of the computational complexity of the proposed algorithm deserve  further research. Moreover, it would be interesting to study the robustness properties of the iteration scheme with respect to process and measurement disturbances.


text:
\begin{document}

\title
[tame dynamical systems]{On tame dynamical systems}

\author[E. Glasner]{E. Glasner}
\address{Department of Mathematics,
Tel-Aviv University, Ramat Aviv, Israel}
\email{glasner@math.tau.ac.il}

\begin{abstract}
A dynamical version of the Bourgain-Fremlin-Talagrand
dichotomy shows that the enveloping semigroup of a
dynamical system is either very large and contains
a topological copy of $\beta \N$, or it is a
``tame" topological space whose topology is determined by
the convergence of sequences. In the latter case we say
that the dynamical system is tame.
We show that (i) a metric distal minimal system is tame iff it is
equicontinuous (ii) for an abelian acting group a tame metric minimal
system is PI (hence a weakly mixing minimal
system is never tame), and (iii) a tame minimal cascade
has zero topological entropy.
We also show that for minimal distal-but-not-equicontinuous systems
the canonical map from the enveloping operator
semigroup onto the Ellis semigroup is never an isomorphism.
This answers a long standing open question.
We give a complete characterization of minimal
systems whose enveloping semigroup is metrizable.
In particular it follows that for abelian acting group
such a system is equicontinuous.
\end{abstract}

\thanks{{\em 2000 Mathematical Subject Classification
54H20}}

\keywords{Enveloping semigroup, Enveloping operator semigroup,
tame, injective dynamical system, minimal system,
equicontinuous system, Rosenthal compact,
Fr\'echet compact.}

\maketitle

\section*{Introduction}

The {\em enveloping} (or {\em Ellis }) semigroup
of a dynamical system was
introduced by R. Ellis in \cite{E1}. It proved to be
an indispensable tool in the abstract theory of topological
dynamical systems (see e.g. Ellis
\cite{E2}). However explicit computations of enveloping
semigroups are quite rare. Some examples are to be found in
Namioka \cite{Na} (1984), Milnes \cite{Mil1} (1986)
and \cite{Mil2} (1989),
Glasner \cite{Gl1} (1976) and \cite{G3}
(1993),
Berg, Gove \& Hadad \cite{BGH} (1998),
Budak, I\c sik, Milnes \& Pym. \cite{BIMP} (2001), and
Glasner \& Megrelishvili \cite{GM} (2004).
Rarely is the enveloping semigroup metrizable
(a notable exception is the case of weakly almost
periodic metric systems; see Downarowicz \cite{Dow} (1998) and Glasner
\cite{Gl2} (2003), Theorem 1.48).

In an interesting paper \cite{Ko}, A. K\"ohler pointed
out the relevance of a theorem of Bourgain, Fremlin \& Talagrand
\cite{BFT} to the study of enveloping
semigroups. She calls a dynamical system, $(X,\phi)$,
where $X$ is a compact Hausdorff space and $\phi: X \to X$
a continuous map, {\em regular} if for every function
$f\in C(X)$ the sequence $\{f\circ \phi^n: n\in \N\}$ does
not contain an $\ell^1$ sub-sequence (the sequence $\{f_n\}_{n=1}^\infty$
is an {\em $\ell^1$ sequence} if there are strictly positive constants
$a$ and $b$ such that
$$
a\sum_{k=1}^n |c_k| \le \left\| \sum_{k=1}^n c_k f_k \right\|
\le b\sum_{k=1}^n |c_k|
$$
for all $n\in \N$ and $c_1,\dots,c_n\in \C$).
Since the word ``regular" is already overused in topological dynamics
I will call such systems {\em tame}.
It turns out that for a metric system $(X,\phi)$
this is the same as the condition that $E(X,\phi)$, the enveloping
semigroup of $(X,\phi)$, be a Rosenthal compact (see \cite{GM}).

In that paper K\"ohler also considers another useful notion, that
of the {\em enveloping operator semigroup\/}.
For a Banach space $K$ and a bounded linear operator
$T: K \to K$ this is defined as
$$
\Ecal(T)={\cls}_{w^*}\{T^n :n \in \N\}.
$$
K\"ohler shows that when $(X,\phi)$ is a dynamical system,
$K= C(X)$, and $T:C(X)^* \to C(X)^*$ is the operator induced
by $\phi$ on the dual space $C(X)^*$, then there is always a
surjective homomorphism of dynamical systems
$$
\Phi: \Ecal(T) \to E(X,\phi).
$$
If we view $M(X)$, the compact space of probability
measures on $X$ equipped with the weak$^*$ topology,
as a subset of $C(X)^*$ with ${\spann}(M(X))=C(X)^*$,
we see that this map $\Phi$ is nothing
but the restriction of an element of $\Ecal(T)$ to the subspace
of Dirac measures $\{\del_x: x\in X\}$.
Theorem 5.3 of \cite{Ko} says that for a tame metric dynamical system
$(X,\phi)$, the map $\Phi$ is an isomorphism of the enveloping
operator semigroup onto the Ellis semigroup. (We will
re-prove this theorem in section 1, Theorem \ref{tame-inj}.)
In this paper I will call a dynamical system $(X,\phi)$
for which $\Phi$ is an isomorphism, an {\em injective system}.
In \cite{Ko} there are several other cases where systems are
shown to be injective and the author raises the question
whether this is always the case.
As she points out this question was posed
earlier by J. S. Pym (see \cite{Py}). In
\cite{Im} S. Immervoll gives an example of
a dynamical system which is not injective.
His example is of the form $(X,H)$ where $X=[0,1]$
is the unit interval and $H$ is an uncountable semigroup
of continuous maps from $X$ to itself.
This leaves the question open for $\Z$ (or $\N$) systems,
for group actions and for minimal systems.

In the present work the setup is that of
a compact (mostly metrizable) $\Ga$ dynamical system $(X,\Ga)$
where $\Ga$ is an arbitrary topological group.
In \cite{GM} we have shown that metrizable weakly almost periodic
(WAP) systems and more generally metrizable hereditarily almost
equicontinuous (HAE) systems are tame.
However, most of the results presented here are concerned with the case
where $(X,\Ga)$ is a {\em minimal\/} dynamical system.
In the first section it is shown that
a tame dynamical system is injective.
This, in conjunction with a theorem of Ellis and an old work
of mine on affine dynamical systems (\cite{G2}), is used to deduce
that a metric distal minimal system is injective iff
it is equicontinuous. It therefore follows that
every metric minimal distal-but-not-equicontinuous system
serves as a counterexample to the question of Pym
and K\"ohler. It is also shown that a tame minimal cascade \XT
has zero topological entropy.
In the second section I show that for abelian $\Ga$
a metric minimal tame system is PI, hence in particular
a minimal weakly mixing $\Ga$-system is never tame.
In the third section I consider the case when \XGa
is minimal and $E=E(X,\Ga)$ is metrizable.
Under these assumptions it is shown that
there is a unique minimal ideal $I$ in $E$, that
the group $K$ of automorphisms of the system $(I,\Ga)$
is compact, and that the quotient dynamical system $(I/K,\Ga)$
is proximal.
If we also assume that $\Ga$ is abelian then
$(X,\Ga)$ is equicontinuous.
In the last section I consider the question: how
big can $E(X,\Ga)$ be in $X^X$.

The reader is refered to the sources \cite{E2},
\cite{Gl1}, \cite{V}, \cite{Bro}, \cite{Au}, \cite{Vr} and
\cite{G-lu}, on the abstract theory of topological dynamics and the
structure theory of minimal dynamical systems including
the notion of PI (proximal-isometric) systems.

The questions treated in this paper arose during the work on
another one, \cite{GM}, which I just finished writing jointly
with Michael Megrelishvili. I owe him much for fruitful discussions
on these subjects. I am also indebted to Benjy Weiss
for helpful conversations; in particular the content of
section 4 was the subject of a conversation over lunch
several years ago.

\br

\section{Tame systems are injective}

Recall that a topological space $K$ is called a {\em Rosenthal compact\/}
\cite{Godefroy} if it is homeomorphic to a pointwise compact
subset of the space $B_1(X)$ of functions of the first Baire class
on a Polish space $X$. All metric compact spaces are Rosenthal. An
example of a separable non-metrizable Rosenthal compact is the
{\em Helly compact} of all
(not only strictly) increasing selfmaps
of $[0,1]$ in the pointwise topology.
Another is the {\em two arrows} space
of Alexandroff and Urysohn (see Engelking \cite{Eng}).

A topological space $K$ is a {\em Fr\'echet space} if for
every $A\subset K$ and every $x\in \ol{A}$ there exists a
sequence $x_n\in A$ with $\lim_{n\to\infty} x_n=x$
(see Engelking \cite{Eng}).
A topological space $K$ is {\em angelic}
if every relatively countably compact subset $A \subset K$
satisfies the properties (i) $A$ is relatively compact and
(ii) for every $x\in \ol{A}$
there exists a sequence $x_n\in A$ with $\lim_{n\to\infty} x_n=x$.
Thus a compact space is angelic iff it is Fr\'echet.
Clearly, $\beta \N$, the Stone-\v{C}ech compactification of the
natural numbers $\N$, cannot be embedded into a Fr\'echet space.

The following theorem is due to Bourgain, Fremlin and Talagrand
\cite[Theorem 3F]{BFT}, generalizing a result of Rosenthal. The
second assertion (BFT dichotomy) is presented as in the book of
Todor\u{c}evi\'{c} \cite{T-b} (see Proposition 1 of section 13).

\begin{thm} \label{BFT}
\begin{enumerate}
\item
Every Rosenthal compact space $K$ is angelic.
\item (BFT dichotomy)
Let $X$ be a Polish space and let $\{f_n\}_{n=1}^\infty \subset
C(X)$ be a sequence of real valued functions which is {\em
pointwise bounded}  (i.e. for each $x\in X$ the sequence
$\{f_n(x)\}_{n=1}^\infty$ is bounded in $\R$).
Let $K$ be the pointwise closure of $\{f_n\}_{n=1}^\infty$
in $\R^X$. Then either $K \subset B_1(X)$ (i.e. $K$ is
Rosenthal compact) or $K$ contains a homeomorphic copy of $\beta\N$.
\end{enumerate}
\end{thm}

The following dynamical BFT dichotomy is derived in \cite{GM}.

\begin{thm}[A dynamical BFT dichotomy]\label{D-BFT}
Let $(X,\Ga)$ be a metric dynamical system and let $E=E(X,\Ga)$
be its enveloping semigroup. We have the following alternative.
Either
\begin{enumerate}
\item
$E$ is a separable Rosenthal compact, hence with cardinality
${\card}{E} \leq 2^{\aleph_0}$; or
\item
the compact space $E$ contains a homeomorphic
copy of $\beta\N$, hence ${\card}{E} = 2^{2^{\aleph_0}}$.
\end{enumerate}
\end{thm}

\begin{defn}
We will say that an enveloping semigroup $E(X,\Ga)$ is {\em tame}
if it is separable and Fr\'echet.
A dynamical system $(X,\Ga)$ is {\em tame} when $E(X,\Ga)$ is tame.
\end{defn}

In these terms Theorem \ref{D-BFT} can be rephrased as
saying that {\em a metric dynamical system $(X,\Ga)$ is either
tame or $E(X,\Ga)$ contains a topological copy of $\beta \N$}.
When \XGa is a metrizable system
the group $\Ga$ is embedded in the Polish
group ${\Homeo}(X)$ of homeomorphisms of $X$
equipped with the topology of uniform convergence.
From this fact it is easy to deduce that the enveloping
semigroup $E(X,\Ga)$ is separable.
If moreover \XGa is tame then
$E=E(X,\Ga)$ is Fr\'echet and every element $p\in E$
is a limit of a sequence of elements of $\Ga$,
$p=\lim_{n\to \infty} \ga_n$.

Examples of tame dynamical systems include
metric minimal equicontinuous systems, almost periodic (WAP) systems
(E. Akin, J. Auslander, and K. Berg \cite{AAB}),
and hereditarily non-sensitive (HNS) systems
(Glasner and Megrelishvili \cite{GM}).

The cardinality distinction between the two cases entails
the first part of the following proposition.

\begin{prop}
\begin{enumerate}
\item
For metric dynamical systems tameness is preserved by taking
\begin{enumerate}
  \item subsystems,
  \item countable self products, and
  \item factors.
\end{enumerate}
\item
Every metric dynamical system \XGa admits a unique maximal
tame factor.
\end{enumerate}
\end{prop}

\begin{proof}
As pointed out, the first statement follows from
cardinality arguments (note that $E(X,\Ga)=E(X^\kappa,\Ga)$
for any cardinal number $\kappa$). To prove the second use Zorn's
lemma, the first part of the theorem, and the fact that
a chain of factors of a metric system is necessarily
countable, to find a maximal tame factor. Then use the first
part again to deduce that such maximal factor is unique.
\end{proof}

As was mentioned in the introduction the following
theorem is due to K\"ohler; our proof though is
different (see also \cite{Gl2}, Lemma 1.49).

\begin{thm}\label{tame-inj}
Let \XGa be a metric tame dynamical system.
Let $M(X)$ denote the compact convex
set of probability measures on $X$ (with the weak$^*$
topology). Then each element $p\in E(X,\Ga)$ defines
an element $p_*\in E(M(X),\Ga)$ and the map
$p \mapsto p_*$ is both a dynamical system and a semigroup
isomorphism of $E(X,\Ga)$ onto $E(M(X),\Ga)$.
\end{thm}

\begin{proof}
Since $E(X,\Ga)$ is Fr\'echet we have for every $p\in E$ a sequence
$\ga_i\to p$ of elements of $\Ga$ converging to
$p$. Now for every $f\in C(X)$ and every
probability measure $\nu\in M(X)$ we get by the
Riesz representation theorem and
Lebesgue's dominated convergence theorem
$$
\ga_i\nu(f)=\nu(f\circ \ga_i)\to \nu(f\circ p):=p_*\nu(f).
$$
Since the Baire class 1 function $f\circ p$ is well defined
and does not depend upon the choice of the convergent
sequence $\ga_i\to p$, this defines the map $p \mapsto p_*$ uniquely.
It is easy to see that this map is an isomorphism of dynamical systems,
whence a semigroup isomorphism. Finally as $\Ga$ is dense in both
enveloping semigroups, it follows that this isomorphism is onto.
\end{proof}

\begin{defn}
We will say that the dynamical system $(X,\Ga)$ is
{\em injective} if the natural map
$E(M(X),\Ga) \to E(X,\Ga)$ is an isomorphism.
\end{defn}

In these terms the previous theorem can be restated as
follows. {\em A tame dynamical system is injective}.
Our next theorem, which relies on \cite{G1},
answers a question of J. S. Pym
and A. K\"{o}hler (see also S. Immervoll \cite{Im}).

\begin{thm}\label{distal}
A minimal distal metric dynamical system is
injective iff it is equicontinuous.
\end{thm}

\begin{proof}
It is well known that when $(X,\Ga)$ is equicontinuous,
$E=E(X,\Ga)$ is a compact topological group and in that case
it is easy to see that $(X,\Ga)$ is injective.
By a theorem of Ellis (see e.g. \cite{E2}),
a system $(X,\Ga)$ is distal iff $E(X,\Ga)$
is a group. Thus, if $(X,\Ga)$ is distal metric and injective
then $E(X,\Ga)=E(M(X),\Ga)$ is a group and it follows that
the dynamical system $(M(X),\Ga)$ is also distal. By Theorem 1.1
of \cite{G1}, the system $(X,\Ga)$ is equicontinuous.
\end{proof}

\begin{cor}\label{cor-distal}
A  minimal distal metric system is tame iff it is equicontinuous.
\end{cor}

\begin{proof}
A metric minimal equicontinuous system is isomorphic to its
own enveloping semigroup. For the other direction observe
that if \XGa is tame then by Theorem \ref{tame-inj} it
is injective hence, by Theorem \ref{distal}, it is
equicontinuous.
\end{proof}

By way of illustration consider, given an irrational
number $\al\in \R$, the minimal distal dynamical $\Z$-system
on the two torus $(T,\T^2)$ given by:
$$
T(x,y)=(x+\al, y+x)\qquad \pmod 1.
$$
Since this system is not equicontinuous Theorem \ref{distal}
and Corollary \ref{cor-distal}
show that it is neither tame nor injective.

\br

The fact that tame systems are injective also yield the result that
metric tame minimal systems have zero topological entropy.
For this we need the following (simplified version of a) theorem of
Blanchard, Glasner, Kolyada and Maass \cite[Theorem 2.3]{BGKM}.
Recall that a pair of
points $\{x,y\} \subseteq X$ is said to be a
{\em Li--Yorke pair\/} if one has simultaneously
$$ \limsup_{n\to
\infty} d(T^nx,T^ny)=\delta >0,\quad \text {{ and }}
\quad \liminf_{n\to
\infty} d(T^nx,T^ny)=0.
$$
In particular a Li--Yorke pair is proximal.
A set $S\subseteq X$ is called {\em scrambled\/} if any pair
of distinct points $\{x,y\}\subseteq S$ is a Li--Yorke pair.
A dynamical system $(X,T)$ is called
{\em chaotic in the sense of Li and Yorke\/} if $X$ contains an
uncountable scrambled set.

\begin{thm}\label{BGKM}
Let $(X,T)$ be a topological dynamical system with $\htop (X,T)>0$. Let
$\mu$ be a $T$-ergodic probability measure with $\supp(\mu)=X$ and
$h_\mu(X,T)>0$.
Then there exists a topologically transitive subsystem $(W,T\times T)$
with $W\subseteq X\times X$, such that for every open
$U\subseteq X$ there exists a
Cantor scrambled set $K\subseteq U$ with
$K\times K\setminus \Del_X\subseteq W_{tr}$,
where $W_{tr}$ is the set of transitive points in $W$.
Thus a dynamical system with positive topological entropy
is chaotic in the sense of Li and Yorke.
\end{thm}

We note that the set $W$ in Theorem \ref{BGKM}
has the following special form.
There exists a measure theoretical weakly mixing
factor map $\pi:(X,\Xcal,\mu,T) \to (Y,\Ycal,\nu,T)$
with a corresponding measure disintegration
$$
\mu = \int_Y \mu_y\, d\nu(y)
$$
having the property that $\mu_y$ is non-atomic for $\nu$-a.e. $y$.
The subsystem $W$ is then given as $W = \supp(\la)$, where
$$
\la = \mu\underset{\nu}{\times}\mu =
\int_Y \mu_y \times \mu_y \, d\nu(y).
$$
Consequently if $X_0\subset X$ is any $\mu$-measurable set
with $\mu(X_0)=1$ then with no loss of generality we can assume that
for $\nu$ almost every $y$ the measure $\mu_y$ satisfies
the condition $\mu_y(X_0)=1$. It then follows that
the Cantor set in Theorem \ref{BGKM} can be
chosen to be a subset of $X_0$.

\begin{thm}\label{entropy}
A minimal metric tame $\Z$ dynamical system \XT has zero
topological entropy.
\end{thm}

\begin{proof}
By the variational principle it suffices to show that
$h_\mu(T)=0$ for every $T$-invariant probability measure
$\mu$ on $X$. Let $\mu$ be such a measure. By Theorem
\ref{tame-inj} \XT is injective and therefore $v_*(\mu)=\mu$
for any minimal idempotent $v\in E=E(X,T)$.
Since $v_*\mu(f)=\mu(f\circ v)$ for every $f\in C(X)$
it follows that $\mu(vX)=1$.
(Note that $vX$ is an analytic set hence universally measurable.)
Now if $h_\mu(T)>0$ then
by Theorem \ref{BGKM}, with $X_0=vX$, there is a
Cantor set $K\subset X_0$ such that for
every $x,x'$ distinct points in $K$
the pair $(x,x')$ is a proximal pair.
However, since pairs $(x,x')\in vX \times vX$
with $x\not= x'$ are {\em almost periodic\/}
(i.e. have minimal orbit closure in $X\times X$)
they are never proximal pairs and we conclude that $h_\mu(T)=0$.
\end{proof}

\begin{rmk}
In the proof of Theorem \ref{entropy}, with slight modifications,
one can use instead of the results in \cite{BGKM} a theorem
of Blanchard, Host and Ruette \cite{BHR} on the abundance of asymptotic
pairs is a system \XT with positive topological entropy.
\end{rmk}

\section{Minimal tame systems are PI}

As we have seen, when \XGa is a metrizable tame system
the enveloping semigroup $E(X,\Ga)$ is a separable Fr\'echet
space. Therefore each element $p\in E$
is a limit of a sequence of elements of $\Ga$,
$p=\lim_{n\to \infty} \ga_n$. It follows that the subset
$C(p)$ of continuity points of each $p\in E$ is a
dense $G_\del$ subset of $X$. More generally, if $Y\subset X$
is any closed subset then the set $C_Y(p)$ of continuity points
of the map $p\rest Y : Y \to X$ is a dense $G_\del$ subset
of $Y$. For an idempotent $v=v^2\in E$ we write
$C_v$ for $C_{\ol{vX}}(v)$.

\begin{lem}\label{Cp}
Let \XGa be a metrizable tame dynamical system,
$E=E(X,\Ga)$ its enveloping semigroup.
\begin{enumerate}
\item
For every $p\in E$ the set $C(p)\subset X$
is a dense $G_\del$ subset of $X$.
\item
For every minimal idempotent $v\in E$, we have
$C_v\subset vX$.
\item
When $\Ga$ is commutative we have $C(v)\subset vX$.
\end{enumerate}
\end{lem}

\begin{proof}
1.\
See the remark above.

2.\
Given $x\in C_v$ choose a sequence $x_n \in vX$ with
$\lim_{n\to\infty} x_n =x$.
We then have $vx=\lim_{n\to\infty} vx_n = \lim_{n\to\infty} x_n =x$,
hence $C_v\subset vX$.

3.\
When $\Ga$ is commutative we have $\ga p =p \ga$ for every
$\ga\in \Ga$ and $p\in E$. In particular the subset
$vX$ is $\Ga$ invariant hence dense in $X$. Thus $\ol{vX}=X$,
hence $C(v) = C_v \subset vX$ by part 2.
\end{proof}

We next proceed to the main theorem of this section.

\begin{defn}
Let \XGa be a dynamical system. We say that a closed
$\Ga$-invariant set $W\subset X\times X$ is a {\em $M$-set}
if it satisfies the conditions
\begin{enumerate}
  \item The system $(W,\Ga)$ is topologically transitive.
  \item The almost periodic points are dense in $W$.
\end{enumerate}
\end{defn}

A theorem of Bronstein asserts that a metric system \XGa
is PI iff every $M$-set in $X\times X$ is minimal
(\cite{Bro}, see also \cite{G-lu}).

\begin{thm}\label{PI}
Let $\Ga$ be a commutative group. Then
any metric tame minimal system \XGa is PI.
\end{thm}

\begin{proof}
We will prove that the Bronstein condition holds; i.e.
that every $M$-set in $X\times X$ is minimal. So let
$W\subset X\times X$ be an $M$-set.
Let $v=v^2$ be some minimal idempotent in $E(X,\Ga)$.
By Theorem \ref{Cp}.3 the set $C(v)$ of continuity points
of the map $v: X \to X$ is a dense $G_\del$ subset of $X$
and moreover $C(v)\subset vX$. Let $U$ be a relatively
open subset of $W$, then there exists a minimal subset
$M\subset W$ with $M\cap U\not =\emptyset$. Let
$\pi_i: M \to X,\ i=1,2$ denote the projection maps.
Because $M$ is minimal we have $\pi_i(M)=X$ and
the map $\pi_i$ is semi-open; i.e.
${\inte}(\pi_i(\ol{V}))\ne \emptyset$ for every nonempty open
subset $V$ of $M$ (see e.g. \cite[Lemma 1.5]{G1.5};
these observations are due to Auslander and Markley).
It follows that the sets $\pi_i^{-1}(C(v)),\ i=1,2$
are dense $G_\del$ subsets of $M$ and therefore so is
the set
$$
(C(v) \times C(v)) \cap M = \pi_1^{-1}(C(v))\cap \pi_2^{-1}(C(v)).
$$
In particular $U \cap (C(v) \times C(v)) \ne \emptyset$
and we conclude that $W_0=(C(v) \times C(v))\cap W$ is a
dense $G_\del$ subset of $W$.

Let $W_{tr}$ be the dense $G_\del$ subset
of transitive points in $W$ and observe that $W_0\cap W_{tr}
\ne\emptyset$. If $(x,x')$ is a point in $W_0\cap W_{tr}$, then
$\ol{\Ga(x,x')}=W$ and since $(x,x') \in (vX)\times (vX)$
it follows that $W$ is minimal.
\end{proof}

\begin{cor}\label{wm}
Let $\Ga$ be a commutative group and \XGa a minimal weakly
mixing metric tame dynamical system, then \XGa is trivial.
\end{cor}

\begin{proof}
A minimal system which is weakly mixing and PI is necessarily
trivial.
\end{proof}

A direct proof of Corollary \ref{wm} that does not
require the PI theory is as follows.

\begin{proof}
Fix a minimal idempotent $u\in E$ and let $C(u)\subset X$ be
the dense $G_\del$ subset of continuity points of $u$.
Fix some $x\in X$; then, by a theorem of Weiss, $P[x]$
the proximal cell of $x$,
is also a dense $G_\del$ subset of $X$
(see \cite[Theorem 1.13]{Gl2}).
Set $A=C(u)\cap P[x]$, then
for $y\in A$ there is a sequence $\ga_j \in \Ga$
such that $\lim_{j\to\infty} \ga_j x = \lim_{j\to\infty} \ga_j y
= x$. By the continuity of $u$ at $x$ we have
\begin{align*}
ux&=u\lim_{j\to\infty} \ga_j x =\lim_{j\to\infty} \ga_j ux\\
&=u\lim_{j\to\infty} \ga_j y =\lim_{j\to\infty} \ga_j uy,
\end{align*}
so that $(ux,uy)\in P$. This implies $ux=uy$ and we conclude
that $ux=uy$ for every $y\in A$. For an arbitrary element
$\ga\in \Ga$, the set $\ga^{-1} A\cap A$ is a residual subset of $X$
and for each $y$ in this set we get
$ux=u \ga y= \ga uy= uy$. Since $\Ga$ is commutative and $(\Ga,X)$ is
minimal we conclude that $\ga z =z$ for every $z\in X$. Thus
$\Ga$ acts trivially on $X$ and the minimality of $(\Ga,X)$
implies that $X$ is a one point space.
\end{proof}

In \cite{GM} there is an example of a minimal tame dynamical
cascade (i.e. a $\Z$-system) on the Cantor set with
an enveloping semigroup which is not metrizable
(in fact $E$ in this example is homeomorphic to
the ``two arrows" space). This system
has the structure of an almost 1-1 (hence proximal)
extension of an irrational rotation on the circle $\T$.
Another such example is in R. Ellis \cite{Ellis}
where the enveloping semigroup of the $SL(2,\R)$ action
on the projective line $\mathbb{P}$ is shown to be tame but not
metrizable. Here the system $(\mathbb{P},SL(2,\R))$ is proximal.
In view of these examples, Corollary \ref{cor-distal}, Theorem
\ref{entropy}, Corollary \ref{wm},
and Theorem \ref{metric} below,

it is reasonable to raise the following question.

\begin{prob}
Is it true that every minimal metrizable tame system \XGa
with an abelian acting group
is a proximal extension of an equicontinuous system?
(Or, for the general acting group, $X$ is proximally equivalent to
a factor of an isometric extension of a proximal system.)
\end{prob}

\br

\section{Metrizable enveloping semigroups}

In this section we consider the case of a minimal
dynamical system for which $E=E(X,\Ga)$ is metrizable.
Of course then $E$ is tame and if $I\subset E$
is a minimal (left) ideal in $E$ then
the dynamical system $(I,\Ga)$ is metric
with $E(I,\Ga) \cong E(X,\Ga)$ so that it is
also tame.

\begin{thm}\label{metric}
Let \XGa be a minimal dynamical system such that $E=E(X,\Ga)$ is
metrizable. Then
\begin{enumerate}
\item
There is a unique minimal ideal $I\subset E=E(X,\Ga)\cong E(I,\Ga)$,
\item
The Polish group $G_U={\Aut}(I,\Ga)$, of automorphisms
of the system $(I,\Ga)$ equipped with the topology of
uniform convergence, is compact.
\item
The quotient dynamical system $(I/K,\Ga)$ is proximal.
\item
The quotient map $\pi:I \to I/K$ is a $K$-extension.
\item
If in addition \XGa is incontractible then $I=K$ and
$\Ga$ acts on $K$ by translations via a continuous
homomorphism $J:\Ga \to K$ with $J(\Ga)$ dense in $K$.
In particular $(I,\Ga)$, and hence also \XGa, is equicontinuous.
\item
If $\Ga$ is commutative then $X=I=K$ and
$K={\cls}J(\Ga)$ is also commutative.
\end{enumerate}
\end{thm}

\begin{proof}
We split the proof into several steps.

1.\
If $I\subset E$ is a minimal left ideal then \XGa is a factor
of the dynamical system $(I,\Ga)$ and the enveloping
semigroup $E(I,\Ga)$ is isomorphic to $E(X,\Ga)$,
where each $p\in E$ is identified with the map
$L_p:E \to E,\ q\mapsto pq$.

2.\
Let $u=u^2$ be a fixed idempotent in $I$ and, as usual
denote $G=uI\subset I$. Then to each $\al \in G$
corresponds an automorphism $\hat\al: I \to I$,
which is defined by $\hat\al (p) =p\al, \forall \, p\in I$.
The map $G \to G_U,\ \al \mapsto \hat\al$ is a surjective algebraic
isomorphism. The inverse map, $G_U \to G\subset I$ is given by
$\hat\al \mapsto \hat\al(u)=u\al=\al$.
Thus $G_U$ acts on $I$ by right multiplication. In the sequel
we will identify $\hat\al$ with $\al$.

3.\
By Lemma \ref{Cp}, for each idempotent $v=v^2\in I$,
the restricted map $v: \ol{vI} \to vI$,\ $q \mapsto vq$
has a dense $G_\del$ subset $C_v\subset \ol{vI}$ of continuity points.
Again by Lemma \ref{Cp} $C_v \subset  vI$.
Since clearly $C_vG \subset C_v$, we get $C_v=vG=vI$.

4.\
If $p\in vI$ then also $p: \ol{vI} \to vI$ and thus its
set of continuity points $C_p$ is also a dense $G_\del$
subset of $\ol{vI}$. Therefore $C_p\cap C_v = C_p \cap vI\ne\emptyset$,
and since $C_pG\subset C_p$ we conclude that $C_p\supset vI$.

5.\
The $G$-dynamical system $(I,G)$ admits a minimal subset $M$,
and it is clearly of the form $M=\ol{vI}=\ol{vG}$ for some $v=v^2\in I$.
By minimality we have $M=\ol{wI}$ for any other idempotent
$w=w^2\in \ol{vI}$. Since, by step 3, $C_v=vI$ and
$C_w=wI$ are residual subsets of $M$, their intersection
is non empty and the structure of $I$ as a disjoint union
of groups implies that $v=w$, hence $wI = vI =M$.
Thus $v: I \to I, \ p\mapsto v p$
has a closed range $vI$ and the right action
of $G$ on $M=vI=vG$ is algebraically transitive.
(The right action of $G$ on $I$,
hence also on $vI$ is free.)
Moreover, from step 4 we see that every $\al\in G$
acts continuously on $vI$ on the {\em left\/}; i.e.
$p_n \to p$, for $p_n,p\in vI$ implies $\al p_n \to \al p$.
Thus in the compact group $vG$, with the topology inherited
from $I$, both left and right multiplications are continuous.
By a theorem of Ellis (\cite{E0}) it follows that $vG$ is a compact
topological group. Being a closed subset of $I$ it is
also Polish.

6.\
Now the map $v: G_U \to vG, \ \al \mapsto v\al$
is clearly a continuous surjective 1-1 homomorphism of Polish topological
groups and a theorem of Banach (\cite{B}) implies
that it is a topological isomorphism (see also \cite[Lemma 3]{G2}).
We therefore conclude that $G_U$ is a compact subgroup of
${\Aut}(I,\Ga)$. Letting $K=G_U$ all the assertions
of the theorem follow readily.
\end{proof}

\begin{rmk}
Let $\Ga$ be a topological group and $J: \Ga \to K$
a continuous homomorphism, where $K$ is a compact metrizable
topological group and $J(\Ga)$ is dense in $K$. In addition let
$H$ be a closed subgroup of $K$ for which
$\bigcap_{\, k\in K} kHk^{-1}=\{e\}$. Then the dynamical system
$(X,\Ga)=(K/H,\Ga)$ where $\ga (kH) = J(\ga)kH,\ (\ga\in \Ga, k\in K)$
is a minimal dynamical system with $E(X,\Ga)= K$.
In fact, these are the only examples I know of minimal systems
with metrizable enveloping semigroup.
\end{rmk}

\begin{prob}
Is there a nontrivial minimal proximal system
with a metrizable enveloping semigroup?
\end{prob}

\br

\section{When is $E(X,\Ga)$ all of $X^X$ ?}

We say that the system $(X,\Ga)$ is {\em $n$-complete}
if for every point $(x_1,\dots,x_n)
\in X^n$ with distinct components
the orbit  $\Ga(x_1,\dots,x_n) $ is dense in $X^n$.
It is called {\em complete\/}
when it is $n$-complete for every $n\in \N$.

\begin{thm}\label{complete}
Let $(X,\Ga)$ be a dynamical system. Then $E(X,\Ga)=X^X$
if and only if $(X,\Ga)$ is complete.
\end{thm}

\begin{proof}
Suppose $E(X,\Ga)=X^X$ and let $(x_1,\dots,x_n),
(x'_1,\dots,x'_n)\in X^n$.
Then there exists an element $p\in E$ with $px_i=x'_i,
\ i=1,\dots ,n$, hence $\ol{\Ga(x_1,\dots,x_n)} = X^n$
and $(X,\Ga)$ is complete.

Conversely if $(X,\Ga)$ is complete then
clearly every element of $X^X$ can be approximated
by an element from $\Ga$. As $E$ is closed this
concludes the proof.
\end{proof}

\begin{cor}
Let $X$ be a topological space which is $n$-homogeneous
for every $n\in \N$ (i.e. the group ${\Homeo}(X)$ acts
$n$-transitively on $X$ for every $n$) then for any
dense subgroup $\Ga\subset {\Homeo}(X)$ the dynamical system
$(X,\Ga)$ is complete, hence $E(X,\Ga)=X^X$.
For example this is the case for the Cantor set $C$, for any sphere
$S^n,\ n\ge 2$, and for the Hilbert cube $Q$.
\end{cor}

If $\phi$ is a nontrivial continuous automorphism of a
system \XGa then $\phi p=p\phi$ for every $p\in E=E(X,\Ga)$.
Thus when the group ${\Aut}(X,\Ga)$ is nontrivial
then $E\subset \{p\in X^X: \phi p=p\phi,\ \forall \phi\in
{\Aut}(X,\Ga)\}$. In particular, when $\Ga$ is commutative
$$
E\subset \{p\in X^X: p \ga=\ga p,\ \forall \ga\in
\Ga\}.
$$
Are there dynamical systems \XGa for which this
inclusion is an equality?
A $\Z$-dynamical system $(X,T)$ is said to have {\em
$2$-fold topological minimal
self-joinings\/} (\cite{J}, \cite{K}) if it satisfies
the following condition.
For every pair $(x,x')\in X \times X$ with $x'\not\in \{T^nx: n\in \Z\}$,
the orbit $\{T^n(x,x'): n\in \Z\}$ is dense in $X\times X$.
If it satisfies the analogous condition for every point
$(x_1,x_2,\dots,x_n)\in X^n$ whose coordinates belong to
$n$ distinct orbits, then \XGa has
{\em $n$-fold topological minimal self-joinings}.
As in the proof of Theorem \ref{complete} it is easy to see that
$$
E(X,T)=\{p\in X^X: p\, T=T p\}
$$
iff \XGa has {\em $n$-fold topological minimal self-joinings\/}
for all $n\ge 1$.
Now in \cite{K}, J. King shows that no non-trivial map has
$4$-fold topological minimal self-joinings. We thus get
the following.

\begin{thm}
There does not exist an infinite minimal cascade \XT for which
$$
E(X,T)=\{p\in X^X: p\, T=T p\}.
$$
\end{thm}

\begin{rmk}
In \cite{W} B. Weiss shows that every aperiodic ergodic zero entropy
measure preserving system has a topological model
which has two-fold topological minimal self-joinings
({\em doubly minimal\/} in his terminology).
\end{rmk}

\bibliographystyle{amsplain}


TITLE: Subadditivity for Renyi entropies
QUESTION [10 upvotes]: Do the Renyi entropies satisfy the standard subadditivity of Shannon entropy? That is,
\begin{equation}
H_\alpha(A,B) \leq H_\alpha(A) + H_\alpha(B)  ?
\end{equation}
for $\alpha \ne 1$. If they do, for which $\alpha$?
Here $H_\alpha(X)$ is the standard Renyi $\alpha$-entropy of the random variable X,
\begin{equation}
H_\alpha(X)=\frac{1}{1-\alpha}\log\sum_i^n p_i^\alpha,
\end{equation}
and $H_\alpha(A,B)$ is the Renyi entropy of the joint probability distribuition of the random variables A and B.

REPLY [2 votes]: In fact, to find a violation of any linear inequality you could form using the Renyi entropies of order $\alpha \in (0,1) \cup (1,\infty)$, please consult the following paper:
http://arxiv.org/abs/1212.0248


TITLE: Find a vector not in the image of a linear transformation
QUESTION [1 upvotes]: The question is Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T(x,y) = (5x+4y, -5x -4y) $ be a linear transformation. Find a vector w that is not in the image of T.
Ok so I have tried a few things:
Firstly I thought well if it is not in the image it is in the kernel. Worked out a basis for the kernel. Answer was wrong.
Then I saw online to find a basis for the image of T which I got to be $(5,-5)$ So any vector in the form a(5,-5) is in the image? But can't a be anything? So don't know how to find the answer.
Then I said Ok well the question is saying
$5x+4y =a$
and
$-5x-4y =b$
So pick an $a $and$ b $ so this is not satisfied. So I tried (1,1) and this is wrong also.
Please help me !!

REPLY [1 votes]: Hint:
Notice that all vectors in the image are of the form $(z, -z)$. Given that, any vector not of that form cannot be in the image. For example ...

REPLY [1 votes]: Note that $(a,b)$ in the image of $T$, say $(a,b)=T(x,y)$ implies $a+b=(5x+4y)+(-5x-4y)=0$. Hence any $(a,b)$ with $a+b\ne 0$ is not in the image, for example $(a,b)=(\pi,1-\sqrt 2)$.


TITLE: Marginal Density Function from Variance
QUESTION [0 upvotes]: I have an assignment for a lesson:

In general, if
E(X2)=[E(X)]2=µ2, determine
fX(x)
I think he made a mistake. E(X2)=[E(X)]2 should be E(X2)-[E(X)]2. Am I right and how can I determine fX(x)

REPLY [2 votes]: Let's rephrase the question.
$X$ is a random variable, and $\mu$ is a constant, such that $\mathsf E(X^2) = \mathsf E(X)^2$ and $\mathsf E(X)^2=\mu^2$.  Determine $f_X(x)$ the probability density function of $X$?
Hint: What does $\mathsf E(X^2) = \mathsf E(X)^2$ mean about the r.v.?


text:
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F

open_locale nat rat real big_operators topological_space

theorem mathd_numbertheory_132 :
  2004 % 12 = 0 :=
begin
  norm_num,
end

subset_name: curated


TITLE: Frequency of Damped Vibrations
QUESTION [1 upvotes]: In the chapter sound, my book states that the Frequency of damped vibrations is less than the natural frequency but I could not understand this because in damped vibrations the amplitude decreases and not the frequency.
May someone please explain this statement to me. I have googled this out but nowhere found the answer to my question rather there were confusing terms like damping coefficients etc. . My book also does not answer the "why" of my question.
P.S. I am a tenth grader so please avoid complicated terminologies.

REPLY [1 votes]: The equation of motion of an undamped spring is simply:
$$ma=-kx$$
Which is usually rewritten as:
$$m\ddot{x}+\omega_0^2x=0$$
With solution:
$$x=A\cos \omega_0t$$
Where the natural frequency $\omega_0$ is:
$$\omega_0=\sqrt{\frac{k}{m}}$$
For the damped case the equation of motion is:
$$ma=-kx-cv$$
Which is usually rewritten as:
$$m\ddot{x}+2\zeta\omega_0\dot{x}+\omega_0^2x=0$$
Where $\zeta$ is the damping ratio:
$$\zeta=\frac{c}{2\sqrt{mk}}$$
In the underdamped area ($\zeta<1$), which is the region of interest to the OP:
$$\omega_1=\omega_0\sqrt{1-\zeta^2}$$
Where $\omega_1$ is the damped frequency.
Obviously $\omega_1>\omega_0$.