# Sensor Noise Rejection

Rakshit Allamraju  Ben Reish  Allan Axelrod

###### Abstract

An inaccessible control architecture caused an undesirable influence on a UAV.
The encountered noise in the performance was modeled using stochastic methods
and a corrective term was implemented on an external controller. Our findings
suggest that the sonar noise problem is unconventional may warrant the
development of a new methodology.

Keywords:  Subsumption Architecture,Gaussian Mixture Models,Gaussian Process,
Quadcopter, Random Variable

##  1 Introduction

Industrial applications of controlled solutions vary widely in the level of
documentation made available to the end-users. In some instances, entire
portions of the software for a controlled solution are designed to be
inaccessible. This limits the usefulness of purchased controlled solutions to
certain niches that may not adequately encompass the scope of an engineering
project [8]. In such a case, projects may be delayed until an entirely new
controlled solution is purchased, shipped, activated and reprogrammed.
However, project delays could result in unacceptable costs; e.g., in emergency
response scenarios, and delays in implementation could result in the
unnecessary loss of human lives [6, 5].

Additionally, the increasingly aggressive nature of cyber-attacks has posed a
serious security and safety concern . In fact, the power, automation and
chemical industries have already been the targets of debilitating cyber-
attacks [3, 4]. The cyber-warfare arms race has evolved in a costly and overly
complicated manner, often involving efforts related to securing a network of
systems with many accessible input sites. Such efforts typically involve a
trade off between operational efficiency and security.

In response to the aforementioned issues, we present an augmented control
architecture that may similarly augment commercial-off-the-shelf (COTS)
hardware and potential cyber-warfare targets. The augmented architecture
enlists an external sensory system to characterize temporally anomalous or
otherwise undesirable behavior to shape the system input signal such that the
undesirable aspects of the system behavior are canceled out. In the context of
COTS systems, this may allow for systems to operate in more exotic
environments and fulfill a broader range of operational capabilities. For
cyber-warefare targets, this solution may allow systems to continue to exhibit
desirable performance before, during and after a cyber-attack

For the sake of simplicity, we experimentally demonstrate the effectiveness of
the augmented control architecture in the context of COTS systems. The
demonstrated noise rejection is notably different than what we’ve seen in
literature and this report is an initial survey of our investigation.

##  2 Problem Formulation

The objective is to characterize and reduce the undesirable input (noise)
introduced into the control input by a lower-level control loop of the
subsumption architecture [1] as the UAV hovers about a location in
3-dimensional space. The UAV has an internal controller that uses a sonar
apparatus to maintain a preset height of $1m$. An Opti-Track system is used as
an external sensor to measure the UAV position during each flight testing
episode. The data from the Opti-Track system is received by a Robot Operating
System (ROS) client, which contains the higher-level control loop of the
subsumption architecture. The ROS client is used to affect the position and
orientation of the UAV. Since the default UAV altitude is above the observable
range of the Opti-Track, the ROS client altitude controller enforces an
alternative steady state altitude of $0.5m$. The composite effect on the UAV
altitude as it hovers over a uniformly level surface is shown in Figure 1a.
The composite effect on the UAV altitude as it hovers over a stepped surface
is shown in Figure 1b. The degradation of the composite UAV altitude control
in the presence of the stepped surface is significant and undesirable.

The approach we have considered for fitting our data are described below.

(a) ZError -No Building (b) ZError -Building

Figure 1: Zerror Comparison

###  2.1 Gaussian Mixture Models

Gaussian mixture models (GMM’s) are one of the most mature approaches in
modeling density estimation of data. GMM’s are parametric probability
distributions which are represented as weighted sums of Gaussian component
densities. GMM’s are employed for multiple purposes such as color-based
tracking and segmentation, classifying color textures in image features [7],
and speaker recognition [9]. A GMM can be represented as

$p(x|\lambda)=\sum\limits_{i=1}^{M}w_{i}g_{i}(x_{i}|\mu_{i},\sigma_{i})$ (1)

GMM’s add a great deal of flexibility for modeling data. If a sufficient
numbers of components is allowed, the data can be easily modeled with a high
degree of accuracy. One of the main abilities of GMM’s is the capability to
form smooth approximations to arbitrarily shaped densities. It can be
considered as a amalgamation of a uni-modal Gaussian model and a vector
quantizer thus enabling us to access characteristics of both. Thus the GMM not
only provides a smooth overall distribution fit which is a feature of uni-
modal Gaussian, but also details the multi-modal nature of the density modeled
by Vector Quantizer.

In our current analysis we sought to use Gaussian Mixture Models to model the
probability distribution in altitude and thus relate it to the noise generated
by the sonar system on the UAV. The presence of the sonar on the controller of
the UAV induces a change in altitude which we try to negate using a direct
inversion controller.

The problem is particularly interesting because the behavior of the sonar
signal is not directly observable, due to the design of the UAV. Because of
this effect we have a deterministic sonar sensor and we need to model it as a
stochastic system to determine its output. The noise introduced by the
presence of a building from the built-in sonar on the UAV is taken as a random
variable. This is analogous to the automotive industry using a Gaussian random
process for characterization of road surfaces [2]. The automotive work
characterized the shock absorber response of simulated quarter-car models as a
random variable [10]. Like the quarter-car model shock absorbers, the UAV does
not have knowledge of upcoming disturbances.

##  3 Experiments and Simulation

In our current experimental setup we use an Opti-Track environment as an
external position measurement sensor and place a UAV in the Opti-Track
environment. The UAV flies in the experimental testbed shown in Figure 2. The
testbed area is $16~{}ft\times 12~{}ft$ and it is equipped with the Opti-Track
motion capture system. The Opti-Track cameras have a optical range between
$18~{}in$ to $433~{}in$, thus allowing an estimated observable volume of
$11~{}ft\times 11~{}ft\times 4~{}ft$. The Opti-Track is a set of 10 infra-red
cameras, which collect data based on the markers attached to the body. The UAV
used in this experiment is the a AR-Drone Parrot 2.0 Quadrotor (UAV), which is
controlled by a client program using the Robot Operating System (ROS) and our
own external controllers. The UAV is designed to receive velocity commands.
The ROS client python script generates the velocity commands and sends the
commands to the UAV. The UAV has two sonar sensors on it’s base at a distance
of $1in$, which are programmed to maintain the UAV at a default height of 1
meter.

Figure 2: Test Bed

A single building is placed at a region close to the center. This region is
chosen in order to maintain enduring visibility of the UAV from multiple
cameras and prevent the Opti-Track from losing its position. The trajectory of
the UAV is designed so that it flies around till it reaches the building and
hovers over it until the end of the episodic run. The data is recorded only
for those intervals when the UAV is hovering over the building.

A Proportional-Derivative (PD) controller written in Python is used as a
closed loop position controller which receives data from the Opti-Track and
calculates the control commands required to maintain the correct position. In
the current setup our desired position in altitude is $0.5m$.

In our initial experiments we use our baseline PD controller to fly the
desired trajectory without the corrective control. Each time the UAV hovers
over the building the sonar detects the loss in altitude and sends a signal to
the internal on-board controller causing it to rise to a higher altitude.
However, this effect pushes the UAV from the building, causing the sonar to
mistake the altitude to be too high and thus sending commands to the on-board
controller to decrease the altitude, leading to a rapid drop in altitude.

We record the position and attitude data and use it to form a distribution of
the position along the Z direction. The distribution along all three
directions are given in Figure 3 a,b,c.

(a) Hist X (b) Hist Y (c) Hist Z

Figure 3: Histogram distributions

##  4 Solution

Our method is to characterize the noise of the sonar and then cancel it out by
updating the desired height. This is an offline method at present, but could
be taken to an online case.

###  4.1 Characterization

We use recorded data to gain an understanding of the noise. The UAV was
initially instructed to hover over a flat (no buildings) surface and the x, y,
and z position of the craft was recorded from the Opti-Track data. The UAV was
then instructed to hover over a disturbance (the edge of a building) and the
x, y, and z position data was again recorded. The data was trimmed to remove
obvious outliers as well as takeoff and landing phenomena.

The gathered z-data was placed in bins according to the corresponding x and y
position. Initially, a polynomial fit of the histogram in Figure 4. was
generated in MATLAB as shown in Figure 5. The intent was to use the polynomial
fit to generate a corrective control term to eliminate the effects of the
sonar. The z-data was also used to train a Gaussian process to locally
estimate the expected altitude of the UAV and fit the data onto a surface.

Figure 4: Z Means in each Bin Figure 5: Meansurface

###  4.2 Cancellation

The estimates generated by the polynomial surface and the surface generated by
the Gaussian Process were not sufficient to eliminate the sonar noise since
the sonar instigated both steep and gradual changes in the altitude of the
UAV; i.e., a single surface-estimate of the UAV altitude could not anticipate
both sonar modes. A heuristic controller solution was also attempted. The
heuristic controller solution resulted in an insufficient improvement in
performance; i.e., the sonar noise was not negated.

The flat terrain data was subtracted from the disturbed data and that result
was used to form the corrective controller command. For a given position in
the x-y space, the desired height command was shifted by the expected value of
the noise from the sonar in the quadcopter’s controller. This allowed the
sonar to add its noise while not adversely affecting the altitude of the
craft.

##  5 Conclusions

We have used a subset of existing tools at our disposal to model the sonar
output in terms of the obtained position data. The random interaction of the
sonar with the stepped surface caused conventional statistical modeling
techniques to generate intractable estimations of the UAV altitude due to the
sonar. Our results suggest that the problem of eliminating sonar noise is
unconventional, and that may warrant the development of a new methodology.
Since the best result was obtained through the heuristic tuning of certain
parameters, it is possible that a reinforcement learning approach might be
more robust.

## References

  * [1] R. Brooks, “A robust layered control system for a mobile robot.”
  * [2] C. J. Dodds and J. D. Robson, “The description of road surface roughness,” _Journal of Sound and Vibration_ , vol. 31, no. 2, pp. 175–183, 1973.
  * [3] D. Kundur, X. Feng, S. Liu, T. Zourntos, and K. L. Butler-Purry, “Towards a framework for cyber attack impact analysis of the electric smart grid,” in _IEEE Smart Grid Comm_. IEEE, October 2010.
  * [4] J. A. Lewis, “Assessing the risks of cyber terrorism, cyber war and other cyber threats,” Center for Strategic and International Studies, Study, 2002. [Online]. Available: www.csis.org
  * [5] R. R. Murphy, “Mobile robots in mine rescue and recovery,” _IEEE Robotics and Automation Magazine_ , vol. 16, pp. 91–103, June 2009.
  * [6] R. R. Murphy and S. Stover, “Rescue robots for mudslides: A descriptive study of the 2005 La Conchita mudslide response,” _Journal of Field Robotics_ , vol. 25, pp. 3–16, Jan 2008.
  * [7] Permuter, “A study of gaussian mixture models of color and texture features for image classification and segmentation.”
  * [8] S. S. R. R. Murphy, J. Kravitz, “Navigational and mission usability in rescue robots,” _Journal of the Robotics Society of Japan_ , vol. 28, pp. 142–146, March 2010.
  * [9] D. Reynolds, “Gaussian mixture models,” MIT Lincoln Laboratory, Tech. Rep., 2009\.
  * [10] G. Verros, S. Natsiavas, and C. Papadimitriou, “Design optimization of quarter-car models with passive and semi-active suspensions under random road excitation,” _Journal of Vibration and Control_ , vol. 11, pp. 581–606, 2005\.


# A compact manifold with infinite-dimensional co-invariant cohomology

Mehdi Nabil  M. Nabil
Cadi Ayyad University, Faculty of Sciences Semlalia, Department of
Mathematics, Marrakesh. Morocco
<EMAIL_ADDRESS>
Abstract. Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on $M$
by diffeomorphisms one can define the $\Gamma$-co-invariant cohomology of $M$
to be the cohomology of the complex
$\Omega_{c}(M)_{\Gamma}=\mathrm{span}\\{\omega-\gamma^{*}\omega,\;\omega\in\Omega_{c}(M),\;\gamma\in\Gamma\\}$.
For a Lie algebra $\mathcal{G}$ acting on the manifold $M$, one defines the
cohomology of $\mathcal{G}$-divergence forms to be the cohomology of the
complex
$\mathcal{C}_{\mathcal{G}}(M)=\mathrm{span}\\{L_{X}\omega,\;\omega\in\Omega_{c}(M),\;X\in\mathcal{G}\\}$.
In this short paper we present a situation where these two cohomologies are
infinite dimensional.

Mathematics Subject Classification 2010: 57S15, 14F40.
Keywords: Cohomology, Transformation Groups.

##  1\. Introduction

In [1], the authors have introduced the concept of co-invariant cohomology. In
basic terms it is the cohomology of a subcomplex of the de Rham complex
generated by the action of a group on a smooth manifold. The authors showed
that under nice enough hypotheses on the nature of the action, there is an
interplay between the de Rham cohomology of the manifold, the cohomology of
invariant forms and the co-invariant cohomology, and this relationship can be
exhibited either by vector space decompositions or through long exact
sequences depending on the case of study (Theorems $1.1$ and $1.3$ in [1]).
Among the various consequences that can be derived from this inspection, it is
evident that the dimension of de Rham cohomology has some control over the
dimension of the co-invariant cohomology, and in most cases presented in [1]
the latter is finite whenever the former is. This occurs for instance in the
case of a finite action on a compact manifold or more generally in the case of
an isometric action on a compact oriented Riemannian manifold, and this fact
holds as well for a non-compact manifold as long as one requires the action to
be free and properly discontinuous with compact orbit space. A concept closely
related to co-invariant cohomology is the cohomology of divergence forms,
which is defined by means of a Lie algebra action on a smooth manifold and was
introduced by A. Abouqateb in [2]. In the course of his study, the author gave
many examples where the cohomology of divergence forms is finite-dimensional.

The goal of this paper is to show that this phenomenon heavily depends on the
nature of the action in play, and that without underlying hypotheses, co-
invariant cohomology and cohomology of divergence forms are not generally
well-behaved. This is illustrated by an example of a vector field action on a
smooth compact manifold giving rise to infinite-dimensional cohomology of
divergence forms and whose discrete flow induces an infinite-dimensional co-
invariant cohomology as opposed to the de Rham cohomology of the manifold.
This shows in particular that many results obtained in [1] and [2] cannot be
easily generalized and brings into perspective the necessity to look for finer
finiteness conditions of co-invariant cohomology in a future study which would
put the present paper in a broader context.

The general outline of the paper is as follows : In the first paragraph, we
briefly recall the notions of co-invariant forms and divergence forms, then we
define an homomorphism of the de Rham complex that is induced by a complete
vector field on the manifold, and which maps divergence forms relative to the
action of the vector field onto the complex of co-invariant differential forms
associated to its discrete flow (see (1) and Proposition 2.0.1). The next
paragraph is concerned with the setting on which our cohomology computations
will take place, it comprises a smooth compact manifold, the $3$-dimensional
hyperbolic torus, which can be obtained as the quotient of a solvable Lie
group by a uniform lattice (the construction given here is that of A. EL
Kacimi in [3]), the Lie algebra action considered is by means of a left-
invariant vector field. We then use a number of results to prove Theorem 3.0.1
which states that the operator defined in (1) is an isomorphism between the
complex of divergence forms and the complex of co-invariant forms, hence
allowing to only consider the cohomology of co-invariant forms for
computation. Finally, the last paragraph is dedicated to the main computation
in which we prove that the discrete flow of the vector field in question on
the hyperbolic torus gives infinite-dimensional co-invariant cohomology.

### Acknowledgement

The author would like to thank Abdelhak Abouqateb for his helpful discussions
and advice concerning this paper.

##  2\. Preliminaries

Let $M$ be a smooth $n$-dimensional manifold and denote $\mathrm{Diff}(M)$ the
group of diffeomorphisms of $M$ and $\Large\raisebox{2.0pt}{$\chi$}(M)$ the
Lie algebra of smooth vector fields on $M$. Let
$\rho:\Gamma\longrightarrow\mathrm{Diff}(M)$ be an action of a group $\Gamma$
on $M$ by diffeomorphisms. For an $r$-form $\omega$ on $M$ and element
$\gamma\in\Gamma$, we denote $\gamma^{*}\omega$ the pull-back of $\omega$ by
the diffeomorphism $\rho(\gamma):M\longrightarrow M$. Let
$\Omega_{c}(M)=\oplus_{p}\Omega_{c}^{p}(M)$ denote the de Rham complex of
forms with compact support on $M$ and put:

$\Omega_{c}^{p}(M)_{\rho}:=\mathrm{span}\\{\omega-\gamma^{*}\omega,\;\gamma\in\Gamma,\;\omega\in\Omega_{c}^{p}(M)\\}.$

Any element of $\Omega_{c}^{p}(M)_{\rho}$ is called a $\rho$-co-invariant or
just a ($\Gamma$-)co-invariant when there is no ambiguity. The graded vector
space $\Omega_{c}(M)_{\rho}:=\oplus_{p}\Omega_{c}^{p}(M)_{\rho}$ is a
differential subcomplex of the de Rham complex $\Omega_{c}(M)$, it is called
the complex of co-invariant differential forms on $M$. When $M$ is compact
this complex is simply denoted $\Omega(M)_{\rho}$. In the case where
$\rho:\mathbb{Z}\longrightarrow M$ is the action induced by a diffeomorphism
$\gamma:M\longrightarrow M$, i.e $\rho(n):=\gamma^{n}$, then we get that:

$\Omega^{p}_{c}(M)=\\{\omega-\gamma^{*}\omega,\;\omega\in\Omega^{p}_{c}(M)\\}.$

Let $\tau:\mathcal{G}\longrightarrow\Large\raisebox{2.0pt}{$\chi$}(M)$ be a
Lie algebra homomorphism and denote $\hat{X}:=\tau(X)$ for any
$X\in\mathcal{G}$ then define:

$\mathcal{C}^{p}_{\tau}(M):=\mathrm{span}\\{L_{\hat{X}}\omega,\;X\in\mathcal{G},\;\omega\in\Omega^{p}_{c}(M)\\}.$

Any element of $\mathcal{C}^{p}_{\tau}(M)$ is called a $\tau$-divergence
$p$-form or simply $\mathcal{G}$-divergence form. The graded vector space
$\mathcal{C}_{\tau}(M):=\oplus_{p}\mathcal{C}^{p}_{\tau}(M)$ is a differential
subcomplex of the de Rham complex. If $X$ is any vector field on $M$, with
corresponding Lie algebra homomorphism
$\tau:\mathbb{R}\longrightarrow\Large\raisebox{2.0pt}{$\chi$}(M)$,
$\tau(1):=X$ then:

$\mathcal{C}^{p}_{\tau}(M)=\mathrm{span}\\{L_{X}\omega,\;\omega\in\Omega^{p}_{c}(M)\\}.$

In what follows, $X\in\Large\raisebox{2.0pt}{$\chi$}(M)$ is a complete vector
field and $\phi:M\times[0,1]\longrightarrow M$ is the flow $\phi^{X}$ of the
vector field $X$ restricted to $M\times[0,1]$. We define the linear operator
$I:\Omega(M)\longrightarrow\Omega(M)$ by the expression:

$I(\eta):=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}\phi^{*}\eta\wedge\mathrm{pr}_{2}^{*}(ds)$
(1)

where
$\Large\raisebox{2.0pt}{$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-6.99998pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-4.68001pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-3.3pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.7pt}}\\!\int_{0}^{1}$}:\Omega^{*}(M\times[0,1])\longrightarrow\Omega^{*-1}(M)$
is the fiberwise integration operator of the trivial bundle
$M\times[0,1]\overset{\mathrm{pr}_{1}}{\longrightarrow}M$ (see [4]) and $ds$
the usual volume form on $[0,1]$.
Let $\tau:\mathbb{R}\longrightarrow\Large\raisebox{2.0pt}{$\chi$}(M)$ be the
Lie algebra homomorphism induced by $X$ and let
$\rho:\mathbb{Z}\longrightarrow\mathrm{Diff}(M)$ be the discrete flow of $X$
i.e the group action given by $\rho(n):=\phi_{n}^{X}$.

######  Proposition 2.0.1.

The operator $I:\Omega(M)\longrightarrow\Omega(M)$ defined by
$\eqref{opcodiv}$ is a differential complex homomorphism i.e $I\circ d=-d\circ
I$. Moreover $I(\mathcal{C}_{\tau}(M))\subset\Omega_{c}(M)_{\rho}$ and the
restriction of $I:\mathcal{C}_{\tau}(M)\longrightarrow\Omega_{c}(M)_{\rho}$ is
surjective.

###### Proof.

Let $\eta\in\Omega(M)$ and denote $\iota_{s}:M\longrightarrow
M\times\\{s\\}\hookrightarrow M\times[0,1]$ be the natural inclusion, then
using Stokes formula for fiberwise integration we get that:

$I(d\eta)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}\phi^{*}(d\eta)\wedge\mathrm{pr}_{2}^{*}(ds)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}d(\phi^{*}\eta\wedge\mathrm{pr}_{2}^{*}(ds))=-dI(\eta)+\big{[}\iota_{s}^{*}(\phi^{*}\eta\wedge\mathrm{pr}_{2}^{*}ds)\big{]}_{0}^{1},$

and since $\iota_{s}^{*}\mathrm{pr}_{2}^{*}(ds)=0$ then $I(d\eta)=-dI(\eta)$.
For the second claim we start by showing that $I(\eta)$ has compact support
whenever $\eta$ does. Indeed assume $\eta\in\Omega_{c}(M)$ and denote
$K:=\mathrm{supp}(\eta)$, next consider the map:

$f:M\times\mathbb{R}\longrightarrow
M,\;\;\;(x,s)\mapsto\phi_{s}^{-1}(x):=\phi(x,-s),$

Then $f$ is continuous and therefore $L:=f(K\times[0,1])$ is compact. For any
$y\in M\setminus L$ and any $s\in[0,1]$ we get that $\phi_{s}(y)\notin K$ and
therefore $(\phi^{*}\eta)_{(y,s)}=0$, this implies that $I(\eta)_{y}=0$. We
conclude that $\mathrm{supp}\;I(\eta)\subset L$ i.e $I(\eta)\in\Omega_{c}(M)$.
From the relation $\mathrm{T}_{\small(x,t)}\phi(0,1)=X_{\phi_{t}(x)}$ one gets
that $\phi^{*}\circ i_{X}=i_{(0,\frac{\partial}{\partial s})}\circ\phi^{*}$
and therefore $\phi^{*}\circ L_{X}=L_{(0,\frac{\partial}{\partial
s})}\circ\phi^{*}$. Moreover we have that

$L_{(0,\frac{\partial}{\partial
s})}\mathrm{pr}_{2}^{*}(ds)=0\;\;\;\text{and}\;\;\;\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.83331pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.90001pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.75pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25pt}}\\!\int_{0}^{1}\circ
i_{(0,\frac{\partial}{\partial s})}=0.$

If we write $\eta=L_{X}\omega$ for some $\omega\in\Omega_{c}(M)$ then we get
that:

$\displaystyle I(L_{X}\omega)$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}\phi^{*}(L_{X}\omega)\wedge\mathrm{pr}_{2}^{*}(ds)$
$\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}L_{(0,\frac{\partial}{\partial
s})}(\phi^{*}\omega)\wedge\mathrm{pr}_{2}^{*}(ds)$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}L_{(0,\frac{\partial}{\partial
s})}(\phi^{*}\omega\wedge\mathrm{pr}_{2}^{*}(ds))$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}d\circ
i_{(0,\frac{\partial}{\partial
s})}(\phi^{*}\omega\wedge\mathrm{pr}_{2}^{*}(ds))+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}i_{(0,\frac{\partial}{\partial
s})}d(\phi^{*}\omega\wedge\mathrm{pr}_{2}^{*}(ds))$ $\displaystyle=$
$\displaystyle
d\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{0}^{1}i_{(0,\frac{\partial}{\partial
s})}(\phi^{*}\omega\wedge\mathrm{pr}_{2}^{*}(ds))\right)+[\iota_{s}^{*}i_{(0,\frac{\partial}{\partial
s})}(\phi^{*}\omega\wedge\mathrm{pr}_{2}^{*}(ds))]_{0}^{1}$ $\displaystyle=$
$\displaystyle[\phi_{s}^{*}\omega]_{0}^{1}$ $\displaystyle=$
$\displaystyle\phi_{1}^{*}\omega-\phi_{0}^{*}\omega$ $\displaystyle=$
$\displaystyle\phi_{1}^{*}\omega-\omega.$

It follows that $I(\mathcal{C}_{\tau}(M))\subset\Omega_{c}(M)_{\rho}$. This
also shows that $I:\mathcal{C}_{\tau}(M)\longrightarrow\Omega_{c}(M)_{\rho}$
is surjective.

######  Remark 2.0.1.

Note that $\phi^{X}$-invariant forms on $M$ are fixed by $I$ i.e if
$\omega\in\Omega(M)$ such that $L_{X}\omega=0$ then $I(\omega)=\omega$.

##  3\. The hyperbolic torus

Consider $A\in\mathrm{SL}(2,\mathbb{Z})$ with $\mathrm{tr}(A)>2$. It is easy
to check that $A=PDP^{-1}$ for some $P\in\mathrm{GL}(2,\mathbb{R})$ and
$D=\mathrm{diag}(\lambda,\lambda^{-1})$. Clearly $\lambda>0$ and $\lambda\neq
1$. Hence it makes sense to set
$D^{t}=\mathrm{diag}(\lambda^{t},\lambda^{-t})$ and define
$A^{t}=PD^{t}P^{-1}$ for any $t\in\mathbb{R}$. Next we define the Lie group
homomorphism:

$\phi:\mathbb{R}\longrightarrow\mathrm{Aut}(\mathbb{R}^{2}),\;\;\;t\mapsto
A^{t}$

The hyperbolic torus $\mathbb{T}^{3}_{A}$ is the smooth manifold defined as
the quotient $\Gamma_{3}\backslash G_{3}$ where
$G_{3}:=\mathbb{R}^{2}\rtimes_{\phi}\mathbb{R}$ and
$\Gamma_{3}:=\mathbb{Z}^{2}\rtimes_{\phi}\mathbb{Z}$. The natural projection
$\mathbb{R}^{2}\rtimes_{\phi}\mathbb{R}\overset{p}{\longrightarrow}\mathbb{R}$
induces a fiber bundle structure
$\mathbb{T}^{3}_{A}\overset{p}{\longrightarrow}\mathbb{S}^{1}$ with fiber type
$\mathbb{T}^{2}$ and $p[x,y,t]=[t]$.

If $(1,a)$ and $(1,b)$ are the eigenvectors of $A$ respectively associated to
the eigenvalues $\lambda$ and $\lambda^{-1}$ then:

$v=(1,a,0),\;\;\;w=(1,b,0)\;\;\;\text{and}\;\;\;e=(0,0,-\log(\lambda)^{-1}),$

forms a basis of $\mathfrak{g}_{3}=\mathrm{Lie}(G_{3})$, and we can check
that:

$[v,w]_{\mathfrak{g}_{3}}=0,\;\;\;[e,v]_{\mathfrak{g}_{3}}=-v,\;\;\;\text{and}\;\;\;[e,w]_{\mathfrak{g}_{3}}=w.$
(2)

Denote $X$, $Y$ and $Z$ the left invariant vector fields on
$\mathbb{R}^{2}\rtimes_{\phi}\mathbb{R}$ associated to $v$, $w$ and $e$
respectively, then $\\{X,Y,Z\\}$ defines a parallelism on
$\mathbb{T}_{3}^{A}$, a direct calculation leads to:

$X=\lambda^{t}\left(\frac{\partial}{\partial x}+a\frac{\partial}{\partial
y}\right),\;\;\;Y=\lambda^{-t}\left(\frac{\partial}{\partial
x}+b\frac{\partial}{\partial
y}\right)\;\;\;\text{and}\;\;\;Z=-\log(\lambda)^{-1}\frac{\partial}{\partial
t}.$ (3)

Now denote $\alpha$, $\beta$ and $\theta$ the dual forms associated to $X$,
$Y$ and $Z$ respectively. It is clear that the vector fields $X$ and $Y$ of
$\mathbb{T}^{3}_{A}$ are tangent to the fibers of the fiber bundle
$\mathbb{T}^{3}_{A}\overset{p}{\longrightarrow}\mathbb{S}^{1}$, and that
$\theta=-(\log\lambda)p^{*}(\sigma)$ where $\sigma$ is the invariant volume
form on $\mathbb{S}^{1}$ satisfying $\int_{\mathbb{S}^{1}}\sigma=1$. Assume in
what follows that the eigenvalue $\lambda$ of $A$ is irrational, then from the
relation:

$A\left(\begin{matrix}1\\\
a\end{matrix}\right)=\lambda\left(\begin{matrix}1\\\ a\end{matrix}\right)$

we deduce that $a\in\mathbb{R}\setminus\mathbb{Q}$. This remark leads to:

######  Proposition 3.0.1.

The orbits of the vector field $X$ defined in (3) are dense in the fibers of
the fiber bundle
$\mathbb{T}^{3}_{A}\overset{p}{\longrightarrow}\mathbb{S}^{1}$. In particular
for any $f\in\mathcal{C}^{\infty}(\mathbb{T}^{3}_{A})$, $X(f)=0$ is equivalent
to $f=p^{*}(\phi)$ for some $\phi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$.

###### Proof.

We shall identify $\mathbb{S}^{1}$ with $\mathbb{R}/\mathbb{Z}$. Fix
$[t]\in\mathbb{S}^{1}$ and consider the diffeomorphism:

$\Phi_{t}:\mathbb{T}^{2}\longrightarrow p^{-1}[t],\;\;\;[x,y]\mapsto[x,y,t].$

Then define the vector field $\hat{X}$ on $\mathbb{T}^{2}$ given by:

$\hat{X}_{[x,y]}=T_{[x,y,t]}\phi_{t}^{-1}(X_{[x,y,t]})=\lambda^{t}\left(\frac{\partial}{\partial
x}+a\frac{\partial}{\partial y}\right).$

Since $a$ is irrational we get that the family $\\{1,a\\}$ is
$\mathbb{Q}$-linearly independent and thus the orbits of $\hat{X}$ are dense
in $\mathbb{T}^{2}$, consequently the orbits of $X$ are dense in $p^{-1}[t]$,
this proves the assertion since $t$ is arbitrary.

The following Lemma is of central importance for the development of this
paragraph and for the computations of the next section:

######  Lemma 3.0.1.

Let $f\in\mathcal{C}^{\infty}(\mathbb{T}^{3}_{A})$ then for every
$s\in\mathbb{R}$ we have the following formula:

$Z\big{(}(\phi_{s}^{X})^{*}(f)\big{)}=-s(\phi_{s}^{X})^{*}\big{(}X(f)\big{)}+(\phi_{s}^{X})^{*}\big{(}Z(f)\big{)}.$
(4)

In particular $Z(\gamma^{*}f)=-X(\gamma^{*}f)+\gamma^{*}(Z(f))$ and
$i_{Z}\circ\gamma^{*}=-\gamma^{*}\circ i_{X}+\gamma^{*}\circ i_{Z}$, where
$\gamma:=\phi_{1}^{X}$.

###### Proof.

For any $(x,y,t)\in\mathbb{R}^{3}$, a straightforward computation gives that:

$\displaystyle Z\big{(}(\phi_{s}^{X})^{*}(f)\big{)}(x,y,t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\log\lambda}d(f\circ\phi_{s}^{X})_{(x,y,t)}(0,0,1)$
$\displaystyle=$
$\displaystyle-\frac{1}{\log\lambda}\frac{d}{du}_{|u=0}(f\circ\phi_{s}^{X})(x,y,t+u)$
$\displaystyle=$
$\displaystyle-\frac{1}{\log\lambda}\frac{d}{du}_{|u=0}f(s\lambda^{t+u}+x,as\lambda^{t+u}+y,t+u)$
$\displaystyle=$
$\displaystyle-\frac{1}{\log\lambda}(df)_{\phi_{s}^{X}(x,y,t)}(s\log(\lambda)\lambda^{t},as\log(\lambda)\lambda^{t},1)$
$\displaystyle=$
$\displaystyle-s(df)_{\phi_{s}^{X}(x,y,t)}(\lambda^{t},a\lambda^{t},0)-\frac{1}{\log\lambda}(df)_{\phi_{s}^{X}(x,y,t)}(0,0,1)$
$\displaystyle=$
$\displaystyle-s(X(f)\circ\phi_{s}^{X})(x,y,t)+(Z(f)\circ\phi_{s}^{X})(x,y,t).$

Which achieves the proof.

######  Corollary 3.0.1.

Let $f\in\mathcal{C}^{\infty}(\mathbb{T}^{3}_{A})$ such that $f=\gamma^{*}f$
with $\gamma:=\phi_{1}^{X}$. Then $X(f)=0$ and consequently $f=p^{*}\psi$ with
$\psi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$.

###### Proof.

Since $f=\gamma^{*}f$ we get that for every $n\in\mathbb{Z}$,
$f=(\gamma^{n})^{*}(f)$ thus the preceding lemma gives that:

$Z(f)=-nX(f)+(\gamma^{n})^{*}(Z(f)).$

Consequently we obtain that for every $n\in\mathbb{Z}$:

$|X(f)|\leq\frac{1}{n}\big{(}\lVert
Z(f)\lVert_{\infty}+\lVert(\gamma^{n})^{*}(Z(f))\lVert_{\infty}\big{)}\leq\frac{2}{n}\lVert
Z(f)\lVert_{\infty},$

which leads to $X(f)=0$ and achieves the proof.

In what follows we denote $M:=\mathbb{T}^{3}_{A}$. It is straightforward to
check that:

$d\alpha=-\alpha\wedge\theta,\;\;\;d\beta=\beta\wedge\theta,\;\;\;d\theta=0.$

and that $L_{X}\alpha=-\theta$ and $L_{X}\beta=L_{X}\theta=0$ thus
$L_{X}(\alpha\wedge\beta\wedge\theta)=0$.
Let $\tau:\mathbb{R}\longrightarrow\Large\raisebox{2.0pt}{$\chi$}(M)$ be the
Lie algebra homomorphism corresponding to the vector field $X$ and
$\rho:\mathbb{Z}\longrightarrow\mathrm{Diff}(M)$ the discrete action generated
by $\gamma:=\phi_{1}^{X}$ where $\phi^{X}$ is the flow of $X$, that is,
$\rho(n)(x)=\phi_{n}^{X}(x)$ for any $n\in\mathbb{Z}$.

######  Theorem 3.0.1.

The homomorphism $I:\mathcal{C}_{\tau}(M)\longrightarrow\Omega(M)_{\rho}$
defined in (1) is an isomorphism.

###### Proof.

In view of Proposition 2.0.1 it only remains to prove that $I$ is injective.
Choose $\eta\in\mathcal{C}_{\tau}(M)$ and write $\eta=L_{X}\omega$ for some
$\omega\in\Omega(M)$. Assume that $I(\eta)=0$, in view of the previous
computation this is equivalent to $\omega=\gamma^{*}\omega$.

If $\eta\in\mathcal{C}_{\tau}^{0}(M)$ then Corollary 3.0.1 gives that
$\omega=p^{*}\phi$ for some $\phi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$,
thus $\eta=0$. On the other hand if $\eta\in\mathcal{C}_{\tau}^{3}(M)$ we can
write $\eta=X(f)\alpha\wedge\beta\wedge\theta$ for some
$f\in\mathcal{C}^{\infty}(M)$ satisfying $f=\gamma^{*}f$ and so by Corollary
3.0.1 we get $\eta=0$.

Now for $\eta\in\mathcal{C}_{\tau}^{1}(M)$ we can write:

$\omega=f\alpha+g\beta+h\theta,\;\;\;\;\;f,g,h\in\mathcal{C}^{\infty}(M).$

Applying $I$ to $L_{X}\alpha=-\theta$ leads to
$\gamma^{*}\alpha=\alpha-\theta$. Moreover since $\theta$ and $\beta$ are
$\phi^{X}$-invariant we get that $\beta=\gamma^{*}\beta$ and
$\theta=\gamma^{*}\theta$ thus:

$\gamma^{*}\omega=(\gamma^{*}f)\alpha+(\gamma^{*}g)\beta+(\gamma^{*}h-\gamma^{*}f)\theta,$

hence $\omega=\gamma^{*}\omega$ is equivalent to $f=\gamma^{*}f$,
$g=\gamma^{*}g$ and $h=\gamma^{*}h-f$. The last relation then implies that
$h-(\gamma^{n})^{*}h=nf$ for all $n\in\mathbb{Z}$ and thus:

$\lVert f\lVert_{\infty}\leq\frac{1}{n}\lVert
h-(\gamma^{n})^{*}h\lVert_{\infty}\leq\frac{2}{n}\lVert
h\lVert_{\infty}\underset{n\rightarrow+\infty}{\longrightarrow}0.$

Therefore $f=0$ and $h=\gamma^{*}h$, $g=\gamma^{*}g$ which according to
Corollary 3.0.1 gives that $X(g)=0$ and $X(h)=0$, and so using that
$L_{X}\beta=0$ and $L_{X}\theta=0$ it follows that $\eta=L_{X}\omega=0$.
Finally let $\eta\in\mathcal{C}^{2}_{\tau}(M)$ and write:

$\omega=f\alpha\wedge\beta+g\alpha\wedge\theta+h\beta\wedge\theta,\;\;\;\;\;f,g,h\in\mathcal{C}^{\infty}(M).$

Then using $\gamma^{*}\alpha=\alpha-\theta$ we obtain that:

$\gamma^{*}\omega=(\gamma^{*}f)\alpha\wedge\beta+(\gamma^{*}g)\alpha\wedge\theta+(\gamma^{*}h+\gamma^{*}f)\beta\wedge\theta,$

and so $\omega=\gamma^{*}\omega$ is equivalent in this case to
$f=\gamma^{*}f$, $g=\gamma^{*}g$ and $h=\gamma^{*}h+\gamma^{*}f$. As before,
this leads to $f=0$, $X(g)=0$ and $X(h)=0$ and so:

$\eta=L_{X}\omega=L_{X}(g\alpha\wedge\theta+h\beta\wedge\theta)=gL_{X}(\alpha\wedge\theta)=-g\theta\wedge\theta=0.$

Thus $I:\mathcal{C}_{\tau}(M)\longrightarrow\Omega(M)_{\rho}$ is an
isomorphism.

This result gives in particular that
$\mathrm{H}(\mathcal{C}_{\tau}(M))\simeq\mathrm{H}(\Omega(M)_{\rho})$ and
therefore we only need to compute the cohomology of $\rho$-co-invariant forms
in this case.

##  4\. Cohomology computation

We now have all the necessary ingredients to perform our computation. Let $M$
denote the hyperbolic torus $\mathbb{T}^{3}_{A}$ defined in the previous
section with $A$ having irrational eigenvalues and let
$X,Y,Z\in\Large\raisebox{2.0pt}{$\chi$}(M)$ be the vector fields defined in
(3) with respective dual $1$-forms $\alpha$, $\beta$ and $\theta$. Define the
action $\rho:\mathbb{Z}\longrightarrow\mathrm{Diff}(M)$ to be the discrete
flow of the vector field $X$ with $\gamma:=\rho(1)$. The main goal is to prove
that first and second co-invariant cohomology groups are infinite-dimension,
however we shall compute the whole cohomology in order get a global picture.

Calculating $\mathrm{H}^{0}(\Omega(M)_{\rho})$: Choose
$f\in\Omega^{0}(M)_{\rho}$ such that $df=0$, then $f$ is a constant function
equal to $g-\gamma^{*}g$ for some $g\in\mathcal{C}^{\infty}(M)$. Consequently
we obtain that:

$\int_{M}f\alpha\wedge\beta\wedge\theta=\int_{M}(g-\gamma^{*}g)\alpha\wedge\beta\wedge\theta=\int_{M}g\alpha\wedge\beta\wedge\theta-\int_{M}\gamma^{*}(g\alpha\wedge\beta\wedge\theta)=0.$

Thus $f=0$ and we conclude that $\mathrm{H}^{0}(\Omega(M)_{\rho})=0$.

Calculating $\mathrm{H}^{1}(\Omega(M)_{\rho})$: We prove that
$\mathrm{H}^{1}(\Omega(M)_{\rho})$ is infinite dimensional. In order to do so,
we prove that the map
$p^{*}:\Omega^{1}(\mathbb{S}^{1})\longrightarrow\mathrm{H}^{1}(\Omega(M)_{\rho})$
is well-defined and injective or equivalently we can show that
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\subset\mathrm{Z}^{1}(\Omega(M)_{\rho})$ and
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\cap\mathrm{B}^{1}(\Omega(M)_{\rho})=0$.

An element $\eta\in p^{*}(\Omega^{1}(\mathbb{S}^{1}))$ can always be written
as $\eta=p^{*}(\phi)\theta$ where
$\phi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$. Since $L_{X}\theta=0$ and
$L_{X}\alpha=-\theta$, then by applying $I$ to $L_{X}\alpha$ we get that
$\theta=\alpha-\gamma^{*}\alpha$, therefore:

$\eta=p^{*}(\phi)\theta=p^{*}(\phi)\alpha-\gamma^{*}(p^{*}(\phi)\alpha).$

Moreover observe that $d\eta=0$, hence we deduce that
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\subset\mathrm{Z}^{1}(\Omega(M)_{\rho})$.
Now suppose $\eta=d(g-\gamma^{*}g)$ then clearly $X(g-\gamma^{*}g)=0$ and
$Z(g-\gamma^{*}g)=p^{*}(\phi)$, thus according to Proposition 3.0.1,
$g-\gamma^{*}g=p^{*}\psi$ for some
$\psi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$. By induction we can show that
for any $n\in\mathbb{N}$, $g=\rho(n)^{*}g+np^{*}\psi$ which then leads to:

$|p^{*}\psi|\leq\frac{1}{n}|g-\rho(n)^{*}g|\leq\frac{2}{n}\lVert
g\lVert_{\infty}\underset{n\rightarrow+\infty}{\longrightarrow}0.$ (5)

Hence $p^{*}\psi=g-\gamma^{*}g=0$ and so $\eta=0$. Thus
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\cap\mathrm{B}^{1}(\Omega(M)_{\mathbb{Z}})=0$.

Calculating $\mathrm{H}^{2}(\Omega(M)_{\rho})$: We will show that
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\wedge\beta\subset\mathrm{H}^{2}(\Omega(M)_{\rho})$.
To do this, we fix a $2$-form $\eta=p^{*}(\phi)\theta\wedge\beta$ such that
$\phi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$. We can easily check that
$d\eta=0$, moreover from the previous calculations and the fact that
$L_{X}\beta=0$ we get that $\beta=\gamma^{*}\beta$, therefore:

$p^{*}(\phi)\theta\wedge\beta=(p^{*}\phi\alpha\wedge\beta)-\gamma^{*}(p^{*}(\phi)\alpha\wedge\beta).$

Hence
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\wedge\beta\subset\mathrm{Z}^{2}(\Omega(M)_{\rho})$.
Now assume that $\eta=d(\omega-\gamma^{*}\omega)$, then using that $[X,Y]=0$
we get $i_{Y}i_{X}(d\omega)=\gamma^{*}(i_{Y}i_{X}d\omega)$ hence according to
Corollary 3.0.1 we can write $i_{X}i_{Y}d\omega=p^{*}\psi$ for some
$\psi\in\mathcal{C}^{\infty}(\mathbb{S}^{1})$. On the other hand we get from
Lemma 3.0.1 that:

$p^{*}\phi-
i_{Y}i_{X}(d\omega)=\gamma^{*}(i_{Z}i_{Y}d\omega)-i_{Z}i_{Y}d\omega.$

It follows from these remarks that
$p^{*}(\phi-\psi)=\gamma^{*}(i_{Z}i_{Y}d\omega)-i_{Z}i_{Y}d\omega$, and as in
(5) we once again prove that $p^{*}(\phi-\psi)=0$, so we deduce that
$p^{*}\phi=p^{*}\psi=i_{Y}i_{X}(d\omega)$. Now if we write
$\omega=f\alpha+g\beta+h\theta$, then we get that $p^{*}\phi=X(g)-Y(f)$.
Moreover, from $X(p^{*}\phi)=Y(p^{*}\phi)=0$ we get that for every
$s\in\mathbb{R}$:

$\displaystyle s^{2}p^{*}\phi$ $\displaystyle=$
$\displaystyle\int_{0}^{s}\int_{0}^{s}(\phi_{t}^{X})^{*}(\phi_{u}^{Y})^{*}(p^{*}\phi)dudt$
$\displaystyle=$
$\displaystyle\int_{0}^{s}\int_{0}^{s}(\phi_{t}^{X})^{*}(\phi_{u}^{Y})^{*}(X(g))dudt-\int_{0}^{s}\int_{0}^{s}(\phi_{t}^{X})^{*}(\phi_{u}^{Y})^{*}(Y(f))dudt$
$\displaystyle=$
$\displaystyle\int_{0}^{s}(\phi_{t}^{X})^{*}X\left(\int_{0}^{s}(\phi_{u}^{Y})^{*}(g)du\right)dt-\int_{0}^{s}(\phi_{u}^{Y})^{*}Y\left(\int_{0}^{s}(\phi_{t}^{X})^{*}(f)dt\right)du$
$\displaystyle=$
$\displaystyle(\phi_{s}^{X})^{*}\left(\int_{0}^{s}(\phi_{u}^{Y})^{*}(g)du\right)-\int_{0}^{s}(\phi_{u}^{Y})^{*}(g)du-(\phi_{s}^{Y})^{*}\left(\int_{0}^{s}(\phi_{t}^{X})^{*}(f)dt\right)+\int_{0}^{s}(\phi_{t}^{X})^{*}(f)dt.$

It follows that:

$s^{2}|p^{*}\phi|\leq
2\left\lVert\int_{0}^{s}(\phi_{u}^{Y})^{*}(g)du\right\lVert_{\infty}+2\left\lVert\int_{0}^{s}(\phi_{t}^{X})^{*}(f)dt\right\lVert_{\infty}\leq
2|s|(\lVert g\lVert_{\infty}+\lVert f\lVert_{\infty})$

Hence $|p^{*}\phi|\leq\dfrac{2}{|s|}(\lVert g\lVert_{\infty}+\lVert
f\lVert_{\infty})\underset{s\rightarrow+\infty}{\longrightarrow}0$.
We conclude that $\eta=0$ and
$p^{*}(\Omega^{1}(\mathbb{S}^{1}))\wedge\beta\cap\mathrm{B}^{2}(\Omega(M)_{\rho})=0$,
in particular this proves that $\mathrm{H}^{2}(\Omega(M)_{\rho})$ is infinite
dimensional.

Calculating $\mathrm{H}^{3}(\Omega(M)_{\rho})$: The elements of
$\Omega^{3}(M)_{\rho}$ are of the form:

$(f-\gamma^{*}f)\alpha\wedge\beta\wedge\theta,$

for some $f\in\mathcal{C}^{\infty}(M)$. Put:

$c=\dfrac{\int_{M}f\alpha\wedge\beta\wedge\theta}{\alpha\wedge\beta\wedge\theta},\;\;\;\text{then}\;\int_{M}(f-c)\alpha\wedge\beta\wedge\theta=0.$

Thus $(f-c)\alpha\wedge\beta\wedge\theta=d\omega$ and since
$L_{X}(\alpha\wedge\beta\wedge\theta)=0$ then
$(\gamma^{*}f-c)\alpha\wedge\beta\wedge\theta=d(\gamma^{*}\omega)$ and
therefore it follows that:

$(f-\gamma^{*}f)\alpha\wedge\beta\wedge\theta=d(\omega-\gamma^{*}\omega),$

i.e $\mathrm{H}^{3}(\Omega(M)_{\rho})=0$.

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##  Appendix A Interviews with healthcare consumers and providers

To validate the general idea of helping readers understand medical research
papers, we interviewed six healthcare consumers and providers. We spoke with
both healthcare consumers with prior experience reading medical research (4
total, referred to as C1–4), and healthcare providers who had discussed
findings from medical papers with their patients (2 total, H1–2). Healthcare
consumers and providers were recruited through our personal and professional
networks and by referral from other interviewees.

These interviews yielded a set of scenarios in which readers turn to the
medical literature. These scenarios motivated the design of our interface and
are offered here to inspire future research to help readers engage with the
medical literature.

Our participants read medical literature because they wanted more information
than they could gather from discussions with their doctor or by consulting
conventional patient-facing resources online. This core motivation manifested
in four cases:

  * •

Learning more about the diagnosis: Participants’ expressed a desire to know
more information than what patient pamphlets or their short doctors’
appointments could give them because they wanted to understand the diagnosis
in greater depth (C1, C3).

  * •

Learning background-specific information Participants sought the medical
literature because their situation was somewhat unique compared to the common
diagnosis (e.g., affecting a different part of their body or at a different
age) (C1, C2).

  * •

Becoming aware of emerging treatment options: Participants mentioned that
having chronic illnesses or those without cures (e.g., severe allergies), had
encouraged them to seek out new clinical trials and trial results as a way of
finding new ways of taking care of themselves. (C1, C4)

  * •

Comparing treatment options: Similar to finding new treatments, participants
described trying to decide between different treatments their doctor
recommended or just wanting to know more about these treatments (e.g., results
from clinical trials or alternative treatments) (C1).

These findings support prior work on motivations in consumer health
information seeking (Sommerhalder et al., 2009) and illustrate the benefits of
open-access medical literature (Zuccala, 2010) as an additional resource for
healthcare consumers to find information important to them. A healthcare
provider we spoke to gave similar insights: their patients sought medical
research papers as a source of information to supplement in-person discussions
with their physician (H1).

Conversations with our participants suggested that paper reading presented
issues such as unfamiliar terminology, assessing relevance, and information
overload. C1 and C3 mentioned that many paper titles were already too complex,
or they needed to learn a lot of new medical vocabulary as they read. C4
described the emotional exhaustion of reading through multiple discouraging
results. C2 mentioned how hard it was to assess if research was trustworthy or
relevant to them. All participants mentioned only being able to engage with
research papers for an hour or two before they were exhausted. To develop a
deeper understanding of how these challenges manifest during reading, we
designed a second formative study where we observed non-experts as they
encountered these challenges when reading medical papers.

##  Appendix B Iterative Design

A total of eight participants (N1-8) used two early prototypes of Paper Plain
in qualitative usability evaluations. Participants were recruited from our
institution, our professional networks, and Upwork. In these evaluations,
participants were given a modified scenario from §3.0.2 and read a paper with
the Paper Plain prototype. These evaluations lasted one hour each.

Overall participants reported that using the Paper Plain prototypes helped
them access important information in a paper (N1–6, 8). Participants said that
the features helped them focus their attention while reading (N4) and gave
them a good overview of the paper (N1 and 3). Participants all expressed
excitement for such a tool existing for their own health information seeking.
The usability evaluations also illustrated important design goals for
effective interactive aids in this reading context, which we integrated into
the design of Paper Plain:

Provide gists on-demand. Plain language is not just useful for helping readers
understand the text; it can also help readers avoid reading an abundance of
dense text. Providing plain language throughout a paper can help readers
choose what not to read. N1 used a prototype with only plain language
answering passages (“Answer gists”) and reported that having only answering
passages simplified was restricting their ability to explore the paper on
their own. N3 also wanted gists for scanning other sections of the paper that
might not have an answering passage, such as specific results sections.

Make guidance both discoverable and unobtrusive. Readers often don’t know
where to look for relevant information in research papers. Navigation that
guides readers to relevant sections can save them time and effort, even if it
reduces some of their autonomy.

The Key Question Index gave an accessible overview of a paper, but
participants often did not notice the sidebar toggle until they had spent
considerable effort understanding the paper. For example, two participants (N1
and N3) missed the button to toggle the Key Question Index sidebar, and only
noticed it later in the session when it was pointed out by a researcher. After
seeing the Key Question Index, N1 mentioned that they wished they had seen it
earlier since it would have provided a helpful high level understanding early
on.

At the same time, the sidebar could be intrusive to some participants. One
participant (N5) reported that the sidebar was distracting and occluded other
typical PDF reader features they wanted to access, such as section outlines.
To balance the goal of providing an intuitive guide without clashing with
readers’ other potential reading strategies, Paper Plain’s final Key Question
Index sidebar was opened when a paper loads but was toggleable to other
sidebars and able to be closed.

Supplement, rather than replace, the text. The text is critical; it is where
readers will find nuanced details that would not be available in summaries or
conventional healthcare consumer materials. Features should make the text more
understandable, not replace it. In addition, NLP systems are imperfect, and a
reader who relies solely on generated content can risk misunderstanding the
actual paper. N1 often double-checked gists with the original text and N4 hid
the gists to read the underlying text. We wanted to make sure that the system
focused readers on the original text and provided generated text as a
supplement, not a replacement. In the prototype the gists were sometimes
overlapping the original text, which made it hard for participants to read
both. In the final design of Paper Plain, all gists were placed as close to
the original text as possible without overlapping. Furthermore, gist content
was provided on-demand, rather than initially displayed along with the paper,
to encourage readers to focus on the paper and pull supplemental content from
the gists only when necessary.

##  Appendix C Paper Plain Implementation

###  C.1. GPT-3 Simplification

We adapted our GPT-3 prompt and generation parameters (e.g., length of
generation and temperature) from one of the preset examples that OpenAI
provides for summarizing text for a 2nd
grader.888https://beta.openai.com/examples/default-summarize We changed the
prompt to summarize for a 5th grader rather than 2nd grader after observing
that using 2nd grader caused the model output to be too general and vague. We
also tested later grades, up to college, but found that the generated text
using the 5th grader prompt was the most consistent. Our final prompt for
GPT-3 was:

> My fifth grader asked me what this passage means: ””” [TEXT TO SIMPLIFY] ”””
> I rephrased it for him, in plain language a fifth grader can understand:

We also updated generation parameters, specifically the length of generation
and temperature (a parameter for controlling the randomness of generations).
We set generation length to 100 characters and temperature to a range of
$0.25$ to $0.5$, depending on the generation.

##  Appendix D Statistical Analysis

###  D.1. Modeling Mixed-Effects in Repeated Measures Studies

For the analysis in § 6.1.5, we used the linear mixed-effects model (LMM).
LMMs are commonly used to analyze data in which the same participant provides
multiple, possibly correlated, measurements, referred to as repeated measures
(Lindstrom and Bates, 1990). LMMs are used as an analysis in medicine (Cnaan
et al., 1997), the behavioral sciences (Cudeck, 1996), and human-computer
interaction (Hearst et al., 2020; Head et al., 2021).

For each of the quantitative measurements discussed in §6.1.3 ($y$), we fit a
LMM with fixed effects $\beta$ for the Paper Plain paper ($x_{1}$) and
interface variant ($x_{2}$) factors.999We also fit the same LMM with an
additional interaction term ($x_{1}x_{2}$) but the $F$-test for this term was
not significant across the three measures ($p>0.67$, $p>0.98$, $p>0.98$). As
such, we proceeded with our analysis without the interaction term in our LMM.
We used the lme4 package in R (Bates et al., 2014) to fit the models. More
precisely, we fit the following LMM:

(1) $E[y]=\beta_{0}+\gamma_{j}+\beta_{1}x_{1}+\beta_{2}x_{2},$

where the random intercepts $\gamma_{j}\sim\mathcal{N}(0,\sigma^{2}_{\gamma})$
capture individual variation of each participant $j$.

We report all the estimated coefficients in Table 3. Due to the categorical
nature of our variables, we interpret the coefficients in the following way:
$\beta_{0}$ is the mean score for Paper Plain while reading the paper for
herniated disc. $\beta_{1}^{SLE}$ is the mean difference in score for the SLE
paper, given the same interface variant. Similarly, $\beta_{2}^{PDF}$,
$\beta_{2}^{SD}$ and $\beta_{2}^{QA}$ are the mean differences in score for
the PDF baseline, Term Definitions and Key Question Index and Answer Gists
interface variants against full Paper Plain variant, given the same paper. For
example, $\beta_{2}^{PDF}=1.9835$ for Reading Difficulty means that the PDF
baseline is associated with a 1.9835 higher difficulty score than Paper Plain,
which is the same result we report in Table 2.

| $\beta_{0}$ | $\beta_{1}^{SLE}$ | $\beta_{2}^{PDF}$ | $\beta_{2}^{SD}$ | $\beta_{2}^{QA}$
---|---|---|---|---|---
Reading Difficulty (1–5) | 2.0884 | 0.3750 | 1.9835 | 1.4851 | 0.3444
Understand (1–5) | 3.8231 | -0.5000 | -1.1769 | -0.7194 | 0.1037
Relevance (1–5) | 3.9316 | -0.5833 | -1.1675 | -0.7524 | 0.1934

Table 3.  Estimated fixed-effect coefficients for the LMM described in
Appendix D for each measurement.

###  D.2. F-Tests for Significant Effect of Interface

We conducted _F_ -tests for differences in fixed-effect estimates between each
interface variant, repeated for each $y$ using the lmerTest R package
(Kuznetsova et al., 2017). Using the Holm-Bonferroni (Holm, 1979) correction
on the $p$-values with the p.adjust R package, we found significance for
reading difficulty $(p<.001)$, relevance $(p<.001)$, and confidence
$(p<.001)$—even while controlling for paper and participant-specific effects.
That is to say, for these metrics, the _F_ -test identified that the choice of
interface (Paper Plain, Questions and Answers, Sections and Terms, or PDF
baseline) is a significant factor. Note that the _F_ -test does not identify
_which_ interfaces differ from one another on the metric.

###  D.3. Tests for Pairwise Differences in Fixed-Effects between Interfaces

To quantify the pairwise differences in fixed-effects between the interface
variants for the measures $y$ under the LMM (and controlling for paper), we
conducted a post-hoc analysis. We used two-sided $t$-tests for pairwise
comparisons using the emmeans R package, yielding the results shown in Table
2.

###  D.4. Ordinal Regression for Likert-Scale Variables

As reading difficulty, confidence, and understanding were measured on a Likert
scale, a LMM estimated means could be ill-suited for analysis, especially if
these measures were not sufficiently normally distributed. We additionally
performed likelihood ratio tests after fitting analogous cumulative link
mixed-effects models (CLMM) provided in the ordinal R package (Christensen,
2018). Likelihood ratio tests, which are similar to _F_ -tests but more
conservative, yielded similar $p$-values—reading difficulty ($p<.001$),
confidence ($p<.001$), and understanding ($p<.001$) —and resulted in the same
conclusions as those when using the LMM. Because pairwise analyses were not
available through emmeans (or other libraries) for CLMMs, we opted to use the
LMM model for these measures to enable subsequent analysis for Table 2.

Source |  Question |  Extracted Answer |  Plain Language Answer
---|---|---|---
PICO |  What condition does this paper study? |  “Systemic lupus erythematosus (SLE) is the prototypical auto-immune connective tissue disease…” |  “Systemic Lupus Erythematosus is a disease that affects about 5 million people in the world…”
|  |  |
PICO |  How is the condition usually treated? |  “Following the diagnosis of SLE, patients are assessed for disease activity and organ involvement, both of which dictate the most appropriate therapy…” |  “After you get the diagnosis of lupus, the doctor will see how bad your lupus is and how much it affects your body…”
|  |  |
Cochrane |  What did the paper want to find out? |  “The aim of this review is to report the evidence concerning the rationale, the efficacy, and the safety of therapeutic peptides…” |  “This is a review of the evidence and reasons why doctors are using peptides to treat lupus…”
|  |  |
Cochrane |  What did the paper do? |  “The next paragraphs report and discuss the current evidence concerning unconjugated and conjugated therapeutic peptides…” |  “The next paragraphs tell us about some drugs that are being tested to see if they can help people with lupus…”
|  |  |
PICO |  What were the new treatment(s), if any this paper looked into? |  “Therapeutic peptides include a class of pharmaceutical com- pounds consisting of amino acid chains of various length (usually less than 40 amino acids)…” |  “A peptide is a small molecule made up of amino acids, which are the building blocks of proteins…”
|  |  |
Cochrane |  What did the paper find? |  “To date, no therapeutic peptide has been licensed and marketed for the use in SLE patients…” |  “A drug that targets a specific part of the immune system is being tested to see if it can help people with a disease called lupus…”
|  |  |
PICO |  Are the findings different depending on a person’s demographics? |  “Being designed on the basis of epitopes that are pathogenic in SLE alone, peptides…” |  “These new drugs are designed to target the bad proteins that cause SLE…”
|  |  |
Cochrane |  What are the limitations of the findings? |  “Nevertheless, despite the successful results observed in preclinical studies, RCTs showed a controversial efficacy profile…” |  “Even though the medicine worked well in the lab, it did not work as well in real life…”
|  |  |
Table 4. Questions used in Key Question Index and Answer Gists. Questions are
presented in the order they appear in the index.