volume+number+eid [cbx@textcite]nameyeardelim

Distance education has a long history. However, COVID-19 has created a new era
of distance education. Due to the increasing demand, various distance learning
solutions have been introduced for different distance education purposes. In
this study, we investigated the impact of COVID-19 on global attention towards
different distance learning-teaching tools. We used Google Trend search
queries as a proxy to quantify the popularity and public interest towards
different distance education solutions. Both visual and analytical approaches
were used to analyse global level web search queries during the COVID-19
pandemic. This can provide a fast first step guide to identifying the most
popular online learning tools available for different educational purposes.
The results allow the teachers to narrow down the search space and deepen
their exploration of prominent distance education solutions to support their
online teaching. The R code and data to reproduce the results of this work are
available in the online supplementary materials.

Keywords: Online learning, Online teaching Distance education solutions,
COVID-19 Pandemic, Google Trend search queries

## 1 Introduction

COVID-19 is still one of the most important priorities of governments and
media in many countries all around the world. Due to the alarming levels of
spread and severity of the virus, the World Health Organization (WHO) declared
the outbreak a Public Health Emergency of International Concern (PHEIC) on
January 30, 2020, and then a pandemic on March 11, 2020 [34]. As of September
29, 2021, more than 232.7 million cases of COVID-19 have been reported in over
192 countries and territories, resulting in more than 4.7 million deaths [13].
In response to the pandemic, health authorities worldwide have taken many
steps, including vaccination development and deployment to minimize the spread
of the virus. In addition, authorities worldwide have also taken many non-
pharmaceutical interventions and preventive measures such as travel
restrictions, lock-downs, workplace hazard controls, school/university
closures, facility closures, and reduction of mass gatherings to reduce the
spread of the virus [7].

Figure 1: Global tracking of COVID-19 caused school closures and re-openings
Figure 2: Region wise tracking of COVID-19 caused school closures and re-
openings

Many sectors have been affected ever since the outbreak of COVID-19,
worldwide. Among these many sectors, education is one of the most affected
sectors with a near-total closure of schools, colleges, and universities all
around the world [12]. Figures 1 show the evolution of global school closures
and reopenings since mid February 2020 [29]. With the start of the pandemic, a
near total closure of schools was observed all around the world. Over time,
Africa and Oceania demonstrated better recovery in comparison to other regions
with their increasing number of fully open schools in which the classes are
held exclusively in person (Figures 2).

Figure 3: Visualization of data from Goolge search trends. (a) COVID-19 cases
worldwide (b) Google search trends of COVID-19 related terms (c) Google search
trends of distance education related terms.

Even though distance education has a long history that goes back almost two
centuries with significant modifications, alterations, and additions in the
process of delivery and communication [22, 26], COVID-19 certainly created a
new era of distance education while encouraging various stakeholders in
education to take the concept of distance education seriously [25]. Sudden
unexpected movement from classrooms to home schooling at large-scale made
students, teachers and parents vulnerable, leading to millions of education
related internet searches being performed during the period of COVID-19
pandemic (Figures 3) [6]. Surprisingly, the search spikes of distance
education related searches (Figure 3 (c)) coincide with increasing COVID-19
counts (Figure 3 (a)) and related internet searches (Figure 3 (b)), even
though distance education has a long history in contrast to the COVID-19
pandemic.

The speed of the physical classroom closures and the rapid move to online
delivery of education allowed little time for planning or reflection on
potential risks that could happen to its various stakeholders, such as
teachers, students, and parents. Teachers were not fully aware of their
obligations and how to connect with students to support their learning. Lack
of preparation, lack of tools and techniques for distance education, lack of
awareness of the availability of the existing tools and techniques and their
effectiveness, lack of interaction and communication with the students, and
lack of awareness of students’ ongoing problems were some of the major issues
they encountered during the COVID-19 pandemic. Students on the other hand who
tend to have fewer educational opportunities beyond schools and universities
are severely affected by this sudden and unexpected movement. Further, the
impact was not limited to students’ learning but also to other aspects of
their lives, such as student debt, digital exclusion, lack of technology, lack
of access to internet-enabled devices or a stable internet connection, long-
term educational disengagement, poor nutrition and food insecurity, increased
psychological challenges, childcare problems, exploitation, school dropouts
and lack of disability services [14, 12, 28, 25, 6]. This also put an extra
burden on parents, especially those with limited education and resources, as
they were expected to facilitate the required learning environment at home.
Working parents also find it difficult to work during a school closure due to
childcare obligations that result from unexpected school closures.
Furthermore, every single step, including planning, developing new tools and
techniques, conducting awareness programmes about distance education, and
shifting towards distance education, happened online, due to the unexpected
massive global shutdown. This situation left the internet as the only medium
to support all these educational processes.

According to [6], for educators to continue to teach inclusively, they need
support to connect with the students through a variety of strategies so that
students feel not only connected to their teachers, but to the subject matter
being taught. In response to this gray situation and the urgent requirement
for a massive transformation from a physical classroom to a virtual learning
environment, UNESCO took immediate and timely action by publishing a long list
of distance learning solutions [30] that can be used to facilitate student
learning during the period of school closure. There are many different
distance learning solutions. Yet very little is known about the uptake of
different solutions or about their effectiveness. Further, getting familiar
with all these available distance learning solutions is also not feasible or
practical due to limited time availability and the need for urgent responses.

Connectivity, flexibility, and the ability to promote varied interactions are
some of the critical aspects educators consider when selecting suitable
distance education solutions. According to [1], the popularity of a product
can also greatly influence consumer purchasing decisions as it can indicate
the prominence of the product in the market, its usability, and its impact.
Further, according to [33], developers also tend to improve their products and
services in response to increasing online search. Therefore, the internet
represents a great opportunity to learn about the popularity and public
attention of a product or service. It is also a way to narrow down the search
space and thereby identify the most suitable products and services that meet
customer needs.

Google Trends is an open source web analytics tool that allows users to get
access to Internet search data that has the ability to provide deep insights
into population behavior [23]. As there is still a struggle as to what
technologies should be used to facilitate student learning, in this study, we
test whether the Google Trends search quiries can be used as a proxy of the
popularity and public interest in different distance education solutions. In
line with that, this paper makes three fundamental contributions to distance
education by exploring three main questions: (1) What is the impact of the
COVID-19 pandemic on education? (2) What solutions are in place to meet the
needs of distance education during COVID-19 pandemic? (3) Which distance
learning solutions have received wide attention and public interest during the
COVID-19 pandemic?

We primarily analyzed quantitative digital footprint data on the internet from
December 2019 to August 2021. The resulted Google Trend footprint provides a
fast first step guide to identifying the most popular distance education tools
available for different educational purposes. It also allows the teachers to
narrow down the search space and deepen their exploration on prominent
distance education solutions to support their online teaching. Furthermore,
identifying the most popular tools and techniques will allow everyone to focus
on those methods and develop a sustainable learning environment by advancing
the existing features.

Quality education is essential to sustainable development and it is a key
United Nations (UN) sustainable development goal. Having the most appropriate
tool is imperative to the successful completion of a task at hand. The
findings of this study provide essential initial guidance to selecting the
most appropriate tool for different distance education needs. Therefore, the
findings of this study directly contribute to the UN’s sustainable development
goals of quality education.

The remainder of this paper is organized as follows: Section 2 presents the
related work to lay the foundation for the Google trend footprint analysis of
different distance learning solutions. Section 3 presents the methodology
followed in the study. Section 4 includes results and discussion. Section 5
concludes the article and presents future research directions.

## 2 Related work

The term distance education is known by a variety of names, such as distance
learning, e-Learning, online learning, home study, independent study, external
study, correspondence education, off-campus study, open learning, and open
education [22]. A few common features found in all these terms are that they
all involve some form of knowledge sharing between two parties; learners and
educators, different times, different geographical locations and/or varying
forms of instructional materials and tools [22]. Further, distance education
has different types of interactions, such as learner–content interactions,
learner–instructor interactions, learner–learner interactions, and
learner–interface interactions that can facilitate learners to construct
knowledge [32]. Different tools are required to fulfill all these requirements
in a distance learning environment.

Google Trends is a freely accessible website sponsored by Google that analyzes
spatio-temporal patterns of search queries in Google Search, across various
regions, subject domains, and languages [5]. Therefore, Google trends provide
a quantifiable and valuable measure of emerging public concerns and trending
topics together with their geographic distribution [2, 11]. According to [18],
public concerns on different matters can also take on an epidemic nature,
starting with the phase of growing interest, which is known as “early
adoption”, to the phase of general interest, which is known as “majority”, and
eventually lose popularity which is known as “lagers stage”. Figure 3 (b) and
(c) also confirm this life cycle explanation of Google Trend search patterns.

Google Trend search queries have become a popular data source in monitoring
and modeling the dynamics of various application domains, such as disaster
management [19], transportation [33], business [8], and health care [23, 2, 3,
5, 11], because they can provide data about social phenomena in a more timely
manner than than traditional data collection processes [31].

According to the study conducted by [33] on taxi preferences, online queries
can provide a proxy for the quality of taxi services and thereby provide a
means to investigate service quality and public concerns about the service
provided. In November, 2008, Google launched an internet-based surveillance
tool, Google Flu Trends (GFT), that uses aggregated Google search data to
provide a near real-time support in predicting influenza outbreaks [11].
Several recent studies have also been conducted to investigate the relation
between Google search trend patterns and the COVID-19 related concerns [17,
15]. The study conducted by [17] focuses on search terms related to the corona
virus, hand washing, and face masks to monitor public restlessness toward
COVID-19. However, there are only a limited number of research attempts
related to education sector using Google Trend search query data [31]. The
study conducted by [31] have used Google Trend search queries to predict
academic fame by testing the correlation between search volume data and
university ranking data. [20] have used a text mining approach to identify
emerging patterns of words and phrases during COVID-19 pandemic, keeping
education prospects as the focus. However, their study was limited to a small
number of learning platforms. According to [22], the design of different types
of learning environments can depend on the learning outcomes, target audience,
access (physical, virtual and/or both), and/or type of content. Therefore,
this study provides a detailed investigation on distance education solutions
focusing on different distance learning needs. In contrast to the work in
[20], the work in our paper has a very specific focus and provides a thorough
investigation on different distance learning solutions available for various
education purposes.

Even though Google Trends search queries provide a massive amount of data,
more care should be given when utilizing it as a research tool as it may
contain inaccuracies [5]. According to [23], the data retrieval process should
be transparent. This will increase the trustworthiness of both the results and
the generalizability of the findings. Furthermore, [23] believe that
researchers should clearly document the rationale and data retrieval process
to ensure the reproducibility of results.

## 3 Methodology

To identify the impact of COVID-19 on education, data on the evolution of
school closures and re-openings was obtained from a repository maintained by
UNESCO [29]. Data related to COVID-19 cases was retrieved from the R software
[24] package coronavirus [21]. The package pulls data from the Johns Hopkins
University Center for Systems Science and Engineering (JHU CCSE) data
repository [13].

In this work, we also used Google Trend search queries as a measurement proxy
to investigate emerging evidence about distance learning and the popularity
and public interest in various distance education solutions. We used weekly
Google Trends search queries of various distance learning solutions during
COVID-19 pandemic. We limited all of our searches to the period from 1
December 2019 to 15 August 2021 to match the period of COVID-19 pandemic.
Through this we wanted to capture the volume of interest in the public realm
about distance education solutions and how this changes over time.

Google Trends determines categories based on search patterns. According to
[31] more relevant and accurate data can be obtained by limiting the search in
Google Trends to a specific category as it helps to reduce noise in the data.
In this work, our focus was given to the education category. Google Trends
does not provide the absolute search volume of a given search term. Instead,
it provides the relative search volume of a particular term, adjusted
according to the total searches of the geography and time range it represents.
The resulted series scales from 0 to 100, where each data point in the series
represents the search interest relative to the highest point in the series for
the selected region and time. [2, 31]. As such, a value of 100 represents the
peak popularity of the term.

It is important to decide what term to search for in Google Trends as
different terms have different search volumes. In the past, several strategies
have been used to select suitable search terms for a given study. Some studies
have selected search terms based on intuition and some through brainstorming
processes [31]. In this study, our choice of search term took into account the
list of distance learning solutions published by UNESCO [30]. Even though, the
solutions they have recommended do not carry their explicit endorsement, they
tend to have a wide reach, a strong user-base, and evidence of impact [30].
For each keyword, the worldwide search queries were performed with the keyword
being used as the “search term”. That allowed us to search for the exact
string of text typed by the user. However, according to [31], short forms or
acronyms had higher search volumes than the corresponding complete names, in
general. However, we did not use the acronym in this study, as our focus is on
specific tools and techniques available in the market to fulfill distance
education needs, and acronyms could be confused with another entity.

We did our analysis under 11 sub-segments namely, digital learning management
systems, systems built for use on basic mobile phones, systems with strong
offline functionality, Massive Open Online Course (MOOC) platforms, self-
directed learning content, mobile reading applications, collaboration
platforms that support live-video communication, tools for teachers to create
digital learning content, external repositories of distance learning
solutions, tools for online proctoring, and resources to provide psychosocial
support. The segmentation was mainly motivated by the list of distance
learning solutions published by UNESCO in response to the COVID-19 pandemic.

In Google trends, it is possible to search for only up to five queries at a
time [31]. Therefore, under each segmentation, we entered up to five distance
learning tools at a time and recorded their relative ranking scores. An
iterative pairwise comparison was first used to identify the series that had
the highest search volume during the study period. Through this iterative
process of relative comparison, we were able to identify the tool with the
highest relative ranking score. Then the relative ranking scores for all the
other tools in the given segmentation were obtained relative to the previously
detected series with the highest ranking scores [31].

Both visual and analytical approaches were used to analyse the changes in web
search queries at the global level to identify the emerging evidence about the
impact of the COVID-19 pandemic on distance education. We also evaluated the
relationship between weekly web searches and the number of global cases of
COVID-19. Cross correlation analysis and the dynamic time warping algorithm
[16] were performed to investigate the relationship and similarities between
the signatures of COVID-19 related search volume data and distance education
related search volume data.

## 4 Results and Discussion

The visual representation of Google Trend search queries of COVID-19 related
terms worldwide shows that the highest peak coincided with the start of the
pandemic (Figures 3 (b)). Surprisingly, the search volume index graph for
distance education related terms also shows a similar spike at the start of
the pandemic (Figures 3 (c)). This highlights the impact of the COVID-19
pandemic on distance education.

According to [22], “distance education” is the most renowned term used when
referencing distance learning. However, both [4] and [10] introduced “online
learning” as the most “recent version,” “newer version” and “improved version”
of distance learning. The Google Trend search patterns correspond to the term
“online learning” confirm the above claim as it is the most searched distance
education related term throughout the COVID-19 pandemic (Figures 3 (c)).
Despite quantitative differences in terms of relative search volume, the
series generated by the two terms “distance learning” and “online teaching”
demonstrate a similar trend during the study period. COVID-19 related web
searches declined in the months after the WHO announcement of the COVID-19
pandemic, despite the increase in COVID-19 cases. A similar decline can be
observed in distance education-related terms (Figures 3). This confirms the
life cycle of Google Trend search patterns introduced in [18]. Further, the
second peak was observed only in education related terms after the WHO
announcement of the COVID-19 pandemic. This could be due to the nearly total
closures of schools that happened with the declaration of COVID-19 as a
pandemic.

Figure 4: Visualization of Dynamic Time Warping Alignments between ’online
learning’ and ’corona’

Figure 5: Visualization of Dynamic Time Warping Alignments ’online learning’
and ’covid’

A pairwise comparison was performed using the Dynamic Time Warping algorithm.
Under this approach, two time series are stretched or compressed locally in
order to make one resemble the other as much as possible. The distance between
the two series is then computed by summing the distances of individual aligned
elements [16]. Two pairwise comparisons, “corona” with “online learning” and
“covid” with “online learning” were conducted to see the impact of COVID-19 on
distance education. Both pairwise comparisons show significant areas of
overlap and a similar dynamic pattern (Figures 4 (a) and 5 (a)). The
normalized distance between the two series in the two comparisons is 0.1326197
and 0.1828299, respectively. This also provides further confirmation of the
similarity of the Google Trend search profiles. This highlights the sudden
public attention towards both aspects, COVID-19 and distance education, even
though distance education has a long history in contrast to COVID-19.

The entirety of the blue warping path flows through green pastures, further
confirming the similarity between the search patterns related to COVID-19 and
distance education (Figures 4 (b) and 5 (b)). The orange volcanic topography
that highlights regions of nonalignment of the series in question if mapped
through this region demonstrates a significant difference in online search
behaviour before and after the COVID-19 pandemic (Figures 4 (b) and 5 (b)).

The degree of digression of the blue line from the ideal 45-degree straight
line indicates a dissimilarity between the two time series (Figures 4 (b) and
5 (b)). In late 2019 the infection was recognized as a virus which is
genetically related to the coronavirus responsible for the Severe Acute
Respiratory Syndrome (SARS) outbreak of 2003, which infected over 8,000 people
from different countries and resulted in more than 700 deaths worldwide [9].
When the virus was known as “corona”, it was not a global concern and was
mostly limited to China. During that time, people did not pay much attention
to online learning as conventional classroom learning approach was the most
popular teaching learning strategy prior to the COVID-19 pandemic. This is
evident from the rough flat line from approximately 120 to 170 on the Query
Index axis in Figure 4 (b).

Later, with the spread of the virus worldwide, WHO named it COVID-19 (COrona
VIrus Diagnosed in 2019) and then declared it a pandemic on March 11, 2020.
With that, a near total closure of schools was observed all around the world
(Figures 1). With this sudden and unexpected massive shutdown, online learning
also became an equally important global concern, similar to the COVID-19
pandemic. This similar behaviour in public attention towards both aspects is
confirmed by the blue line that falls on a near 45-degree line from
approximately 120 to 150 on the Query Index axis in Figure 5 (b). This also
confirms the impact of COVID-19 on distance education and online learning.

Next, a cross correlation analysis was performed to estimate the lag time
between COVID-19 related terms and the distance education related terms
(Figures 6). A close relationship was evident between COVID-19 related search
patterns and the search patterns for the terms “Online learning”, “Distance
learning” and “Online teaching”. The significant cross-correlation
coefficients at lag zero confirm the contemporaneous relationship between the
COVID-19 related search terms and distance learning related search terms. The
significant cross correlation values at lag one or two also confirm the sudden
impact of the COVID-19 pandemic on distance education related public concerns
(Figures 6 (a) - (c) and (f) - (h)). However, no significant synchronization
was observed between the COVID-19 related search trends and the search
patterns of “distance education” or “online proctoring” (Figures 6 (d), (i),
(e), (j)).

Figure 6: Cross-correlation analysis between distance learning related online
search terms and COVID-19 relate online search terms during COVID-19 pandemic

The list of distance education solutions published by UNESCO provides many
different distance education solutions for different educational purposes.
Figure 7 and Figure 8 provide a Google trend footprint that can be used as a
proxy for measuring global attention towards different distance learning
solutions. This provides a fast first step guide to identify the most popular
tools available for different educational purposes. In each panel, one or two
tools have received significantly more attention than the remaining options in
the same category (Figure 7 (a) - (e) and Figure 8 (f) - (j)). However, the
Google Trend series across multiple panels is not comparable as they represent
“relative” search volume for given terms. They are comparable only within a
given panel. The value of 100 is also not comparable across multiple panels
and indicates the maximum popularity for the term within each panel.

Under digital learning management systems “Google Classroom”, which helps
teachers and students connect remotely, communicate, and stay-organized, has
gotten a wide attention and public interest during the COVID-19 pandemic
(Figure 7 (a)). This might be due to its easy and fast setup and usage [27].
Under systems built for use on basic mobile phones, both ‘kaiOS’ and ‘Ubongo’
have received wide attention during the pandemic (Figure 7 (b)). ‘KaiOs’ has
the ability to provide smartphone capabilities to inexpensive, affordable
devices and thereby help open portals to learning opportunities. ‘Ubongo’ has
also received considerable attention, even though its target groups are
school-age children and their parents in Africa. It facilitates localized,
multi-platform entertainment-education to African families at a low cost and
on a massive scale, and is available in both Kiswahili and English. A
significant peak in ‘Ubongo’ series can be observed during the 2020 year end
academic break in Africa (Figure 2). Under systems with strong offline
functionality, ‘Kolibri’, which is available in more than 20 languages, and
supports universal education, has gotten considerable attention during the
COVID-19 pandemic (Figure 7 (c)) as a tool focuses on facilitating learners
living in underserved contexts where the internet is costly, unreliable, or
unreachable. These distance learning solutions can provide a huge relief for
financially distressed families during the COVID-19 pandemic. Under Massive
Open Online Course (MOOC) platforms, ‘Canvas’, which provides free access for
teachers to facilitate learning and professional development, has attracted
considerable attention in comparison to other options available (Figure 7
(d)). ‘Udemy’ has got the second highest attention. However, ‘Canvas’ has
received far more attention than ‘Udemy’. Under self-directed learning
content, ‘Quizlet’ has received considerably high attention throughout the
pandemic in comparison to the other options available under the same category
(Figure 7 (e)). It provides learning flashcards, an AI based learning
assistant, and games to support student learning in multiple subjects and
languages. The second highest option only provides support for language
learning. Quizlet’s ability to provide support for multiple subjects might be
the reason for its ability to outperform its competitors in the market. Under
mobile reading applications, ‘Reads’, which provides digital stories with
illustrations in multiple languages has consistently been the most popular
option (Figure 8 (f)). Under collaboration platforms that support live-video
communication, both ‘WhatsApp’ and ‘Zoom’ had received a lot of attention at
the start of the pandemic in March 2020 (Figure 8 (g)). However, ‘WhatsApp’
has outperformed ‘Zoom’ in attracting public attention during the pandemic. In
contrast, global attention towards ‘Zoom’ has decreased over time. ‘Teams’ has
also gotten considerable attention in comparison to the other remaining
options available in the market. During the pandemic, tools for teachers to
create digital learning content, such as “Trello,” which provides a visual
collaboration platform for teachers for coursework planning, classroom
organization, and collaboration, received a lot of attention. ‘Nearpod’, which
promotes active learning and student engagement through informative and
interactive learning activities has also gotten considerable attention during
the pandemic (Figure 8 (h)). A significant peak can be observed in the
‘Nearpod’ series, during the first academic break. Under external repositories
of distance learning solutions, both ‘Brookings’ and ‘UNHCR’ have got
significant global attention (Figure 8 (i)). ‘Brookings’ provides a large
catalogue of learning innovations, though its focus is not limited to distance
learning solutions. ‘UNHCR’ also provides an extensive list of distance
learning solutions from the United Nations agency for refugees. Student
assessment is vital in ensuring the quality of education. However, due to an
unexpected and sudden school closure, administering examinations at a distance
became a major concern. Under tools for online proctoring, ‘Pearson VUE’ which
provides computer based testing services in over 150 countries worldwide with
more than 5,000 authorized test centers has attracted considerably high public
attention during the pandemic (Figure 8 (j)).

At the start of the pandemic in March 2020, a significant peak of public
attention can be observed in both digital learning management systems (Figure
7 (a)) and collaboration platforms that support live-video communication
(Figure 8 (g)). These peaks align with the near-total closures of schools due
to the COVID-19 pandemic (Figure 1). The major concerns that arise with the
unexpected, sudden movement from a physical classroom to a virtual learning
environment, such as a lack of experience in connecting and sharing learning
materials with students could be the reason for this notable public attention
towards these two aspects. Further, a significant peak can be observed in
digital learning management systems (Figure 7 (a)), Massive Open Online Course
(MOOC) Platforms (Figure 7 (d)), self-directed learning content (Figure 7
(e)), collaboration platforms that support live-video communication (Figure 8
(g)) and tools for teachers to create digital learning content (Figure 8 (h)),
immediately after the academic break in August, 2020 (Figure 1). The inability
to reopen schools fully due to the increasing COVID-19 cases worldwide and the
need for urgent responses to connect with students to somehow facilitate their
learning could be the reasons for this considerable public attention around
August, 2020.

Figure 7: Google Trend footprint analysis of distance learning solutions
during COVID-19 pandemic Figure 8: Google Trend footprint analysis of distance
learning solutions during COVID-19 pandemic Figure 9: Goolge search trend of
the term psychosocial support under All categories and Education category
during COVID-19 pandemic

Good psychosocial support is also vital to improving both students’ and
teachers’ mental health and well-being during a pandemic. This is evident from
the two sudden spikes in the search volume index graphs for the term
‘Psychosocial support’ ( Figure 9 ) which correspond well with the academic
breaks in August in both years, 2020 and 2021, under ‘Education’ category.
This highlights the urgent need for attention and solutions to these aspects.
This type of analysis based on secondary data is vital as these aspects can be
easily overlooked due to the lack of connection and communication between
students and educators during school closures.

## 5 Conclusions and Further work

In this study, we investigated whether the Google Trend search queries can be
used as a proxy of the popularity and public interest in different distance
education solutions. This study is the first detailed analysis of web search
behaviour related to distance learning solutions during the COVID-19 pandemic,
both in quantitative and qualitative terms. A few previous published studies
[31, 20] looked at Google trend search queries related to education, but only
with a limited number of search terms or with different focuses.

Our findings suggest that web search interest in distance education at the
global level was strongly influenced by the COVID-19 pandemic. Google Trend
search queries showed a strong cross correlation between COVID-19 related
search terms and distance education related search terms. The high popularity
and increasing public attention towards different distance learning solutions
during the COVID-19 pandemic was also investigated through a detailed Google
Trend footprint analysis. The findings will help educators to narrow down
their search space and thereby select prominent options available in the
market for different educational purposes. According to [32], the learning
tools can have a significant impact on how students engage in online learning.
The finding will also be helpful for developers to identify their competitors
in the market. The high costs involved with different distance learning
solutions, lack of financial support/funding, and lack of understanding about
the existing distance learning solutions are some of the major issues
educational institutions face with the COVID-19 pandemic. Therefore, the
findings of this study offer important insights to various stakeholders in
education to make better decisions with less time and effort.

Google Trends only captures the search behaviour of a sub-population with
Internet access, even though it is the most common worldwide search engine.
Furthermore, our study focused on global attention toward distance learning
solutions during the COVID-19 pandemic. Therefore, care should be given when
generalizing these findings to different regions or different time periods, as
different regions can have different market orientation and seasonal
requirements. This can lead to variations in the search output and in turn the
study findings for different regions and time frames. Furthermore, Google
Trends data is not available as absolute values but only in the form of
relative volume. Therefore, the scores range from 0 to 100, which does not
allow the researchers to make meaningful comparisons across different
education needs. Further investigation is needed at national and regional
level to identify the specific measures that can be taken to ensure the
quality of distance education at national and regional level.

## Supplementary Materials

Data and scripts: Datasets and R code to reproduce all figures in this article
(https://github.com/pridiltal/Covid19_Online_Teaching_paper).

## Acknowledgment

This work was carried out with the aid of a grant from UNESCO and the
International Development Research Centre, Ottawa, Canada. The views expressed
herein do not necessarily represent those of UNESCO, IDRC or its Board of
Governors.

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\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta-100}
\caption{\it ${\tilde{\b}_\text{eff}} = -100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta4}
\caption{\it ${\tilde{\b}_\text{eff}} = 4$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta100}
\caption{\it ${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta300}
\caption{\it ${\tilde{\b}_\text{eff}} = 300$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta1992-59}
\caption{\it ${\tilde{\b}_\text{eff}} = 1992.59$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H1000/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = 1000$, $GN^2H^2 = 1000$.}
\label{dS_alpha1000_H1000}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta-100}
\caption{\it ${\tilde{\b}_\text{eff}} = -100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta5}
\caption{\it ${\tilde{\b}_\text{eff}} = 5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta100}
\caption{\it ${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta300}
\caption{\it ${\tilde{\b}_\text{eff}} = 300$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta1996-66}
\caption{\it ${\tilde{\b}_\text{eff}} = 1996.66$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = 1000$, $GN^2H^2 = 2 \pi $.}
\label{dS_alpha1000_H2pi}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H0.01/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H0.01/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H0.01/beta10}
\caption{\it ${\tilde{\b}_\text{eff}} = 5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H0.01/beta100}
\caption{\it ${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H2pi/beta300}
\caption{\it ${\tilde{\b}_\text{eff}} = 300$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H0.01/beta1375-79}
\caption{\it ${\tilde{\b}_\text{eff}} = 1375.79$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha1000_H0.01/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = 1000$, $GN^2H^2 = 0.01 $.}
\label{dS_alpha1000_H0.01}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta-4-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta-4-15294}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.15294$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta-2-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -2.7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta-1-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -1.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta20-5646}
\caption{\it ${\tilde{\b}_\text{eff}} = 20.5646$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta50}
\caption{\it ${\tilde{\b}_\text{eff}} = 50$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta48667-7}
\caption{\it ${\tilde{\b}_\text{eff}} = 48667.7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha100_H0.001/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = 100$, $GN^2H^2 = 0.001 $.}
\label{dS_alpha100_H0.001}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta-6}
\caption{\it ${\tilde{\b}_\text{eff}} = -6$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta-5-33788}
\caption{\it ${\tilde{\b}_\text{eff}} = -5.33788$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta-4-96}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.96$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta-4-75}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.75$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta-4-05}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.05$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta0-35}
\caption{\it ${\tilde{\b}_\text{eff}} = 0.35$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta17-4347}
\caption{\it ${\tilde{\b}_\text{eff}} = 17.4347$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H2pi/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = -1$, $GN^2H^2 = 2\pi $.}
\label{dS_alpha-1_H2pi}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_snapshots_H4pi_alpha0/beta-100}
\caption{\it \it${\tilde{\b}_\text{eff}} = -100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_snapshots_H4pi_alpha0/beta-5}
\caption{\it \it${\tilde{\b}_\text{eff}} = -5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_snapshots_H4pi_alpha0/beta-3-53102}
\caption{\it \it${\tilde{\b}_\text{eff}} = -3.53102$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_snapshots_H4pi_alpha0/beta-2}
\caption{\it \it${\tilde{\b}_\text{eff}} = -2$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_snapshots_H4pi_alpha0/beta100}
\caption{\it \it${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\caption{\it $\a = 0$, $GN^2H^2 = 4\pi$.}
\label{dS_H4pi_alpha0}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta-9-41404}
\caption{\it ${\tilde{\b}_\text{eff}} = -9.41404$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta-9-4}
\caption{\it ${\tilde{\b}_\text{eff}} = -9.4$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta-9-25}
\caption{\it ${\tilde{\b}_\text{eff}} = -9.25$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta0}
\caption{\it ${\tilde{\b}_\text{eff}} = 0$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta4-4}
\caption{\it ${\tilde{\b}_\text{eff}} = 4.4$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha-1_H1000/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = -1$, $GN^2H^2 = 1000 $.}
\label{dS_alpha-1_H1000}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha0_H1000/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha0_H1000/beta-11}
\caption{\it ${\tilde{\b}_\text{eff}} = -11$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha0_H1000/beta-7-41404}
\caption{\it ${\tilde{\b}_\text{eff}} = -7.41404$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha0_H1000/beta0}
\caption{\it ${\tilde{\b}_\text{eff}} = 0$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/dS_alpha0_H1000/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it dS, $\tilde{\a} = 0$, $GN^2H^2 = 1000 $.}
\label{dS_alpha0_H1000}
\end{figure}

\FloatBarrier
\subsection{More snapshots for AdS}

As we did for dS in the previous subsection, we provide snapshots for some examples of parameters $(\tilde{\a},GN^2H^2)$ while varying $\tbe$. They are summarized in the following table.
\\

\begin{tabular}{|c|c|c|c|}
\hline
\diagbox{$\tilde{\a}$}{$GN^2\chi^2$} & $<<1$ & $\sim \pi$ & $>>1$ \\ \hline
$-\tilde{\a} >>1$ & \colorbox{blue!30}{\textcolor{green}{Fig.} \ref{AdS_alpha-100_chi0.001}, \textcolor{red}{Fig.} \ref{AdS_alpha-1000_chi0.01}} & \textcolor{red}{Fig. \ref{AdS_alpha-1000_chi2pi} }& \textcolor{red}{ Fig. \ref{AdS_alpha-1000_chi1000}}\\ \hline
$|\tilde{\a}| \leq 2 $ & \textcolor{green}{Fig. \ref{AdS_H0.01_alpha0}} & \colorbox{blue!30}{ \textcolor{green}{Fig.} \ref{AdS_alpha0_chi2pi} , \textcolor{red}{Fig.} \ref{AdS_chi2pi_am2}} &\colorbox{blue!30}{\textcolor{green}{Fig.} \ref{AdS_alpha0_chi1000}, \textcolor{red}{Fig.} \ref{AdS_alpha-1_chi1000}}\\ \hline
$\tilde{\a} >>1$ & \textcolor{green}{ Fig. \ref{AdS_alpha1000_chi0.01}} & \textcolor{green}{Fig. \ref{AdS_alpha1000_chi2pi} }& \textcolor{green}{Fig. \ref{AdS_alpha-1000_chi1000} } \\ \hline
\end{tabular}
\\

$a$ (\ref{sc3}) is either \textcolor{green}{positive} if ``Fig" is written in green, or \textcolor{red}{negative} if it is written in red. \colorbox{blue!30}{Blue boxes} correspond to regimes where the sign of $a$ must be determined by the inequality (\ref{sc5}) :
\be
a<0\iff {\pi\over GN^2\chi^2}<- \left(\tilde{\a} + {1\over 2} \right)
\ee

The conclusion is that a given point in the $(\tilde{\a},GN^2\chi^2)$ plane corresponds either to the behaviour of
\begin{itemize}
\item \textbf{Type A}: Figure \ref{AdS_H0.01_alpha0}, where two tachyons merge on the imaginary axis and form a pair of complex conjugate poles when $\tbe$ is increased.
\item \textbf{Type B}: Figure \ref{AdS_chipiover4_am10}, where all the poles are real valued $\n^2$ for any value of $\tbe$ (no complex pole).
\end{itemize}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta-5-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -5.7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta-5-61689}
\caption{\it ${\tilde{\b}_\text{eff}} = -5.61689$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta-3-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -3.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta-3}
\caption{\it ${\tilde{\b}_\text{eff}} = -3$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta0}
\caption{\it ${\tilde{\b}_\text{eff}} = 0$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta60}
\caption{\it ${\tilde{\b}_\text{eff}} = 60$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta2631-92}
\caption{\it ${\tilde{\b}_\text{eff}} = 2631.92$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi0.01/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}

\caption{\it AdS, $\tilde{\a} = 1000$, $GN^2\chi^2 = 0.01$.}
\label{AdS_alpha1000_chi0.01}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta-20}
\caption{\it ${\tilde{\b}_\text{eff}} = -20$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta-12}
\caption{\it ${\tilde{\b}_\text{eff}} = -12$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta-11-7874}
\caption{\it ${\tilde{\b}_\text{eff}} = -11.7874$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta-9}
\caption{\it ${\tilde{\b}_\text{eff}} = -9$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta-2}
\caption{\it ${\tilde{\b}_\text{eff}} = -2$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta7}
\caption{\it ${\tilde{\b}_\text{eff}} = 7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta1998-16}
\caption{\it ${\tilde{\b}_\text{eff}} = 1998.16$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi2pi/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it AdS, $\tilde{\a} = 1000$, $GN^2\chi^2 = 2 \pi$.}
\label{AdS_alpha1000_chi2pi}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta-17}
\caption{\it ${\tilde{\b}_\text{eff}} = -17$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta-16-8567}
\caption{\it ${\tilde{\b}_\text{eff}} = -16.8567$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta-15}
\caption{\it ${\tilde{\b}_\text{eff}} = -15$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta-14}
\caption{\it ${\tilde{\b}_\text{eff}} = -14$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta50}
\caption{\it ${\tilde{\b}_\text{eff}} = 50$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta1992-1}
\caption{\it ${\tilde{\b}_\text{eff}} = 1992.1$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha1000_chi_1000/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it AdS, $\tilde{\a} = 1000$, $GN^2\chi^2 = 1000$.}
\label{AdS_alpha1000_chi1000}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta-300}
\caption{\it ${\tilde{\b}_\text{eff}} = -300$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta-70}
\caption{\it ${\tilde{\b}_\text{eff}} = -70$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta40}
\caption{\it ${\tilde{\b}_\text{eff}} = 40$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta500}
\caption{\it ${\tilde{\b}_\text{eff}} = 500$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta15988-3}
\caption{\it ${\tilde{\b}_\text{eff}} = 15988.3$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi_1000/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it AdS, $\tilde{\a} = -1000$, $GN^2\chi^2 = 1000$.}
\label{AdS_alpha-1000_chi1000}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta-100}
\caption{\it ${\tilde{\b}_\text{eff}} = -100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta5}
\caption{\it ${\tilde{\b}_\text{eff}} = 5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta100}
\caption{\it ${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta300}
\caption{\it ${\tilde{\b}_\text{eff}} = 300$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta15985-4}
\caption{\it ${\tilde{\b}_\text{eff}} = 15985.4$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi2pi/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it AdS, $\tilde{\a} = -1000$, $GN^2\chi^2 = 2\pi$.}
\label{AdS_alpha-1000_chi2pi}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta-100}
\caption{\it ${\tilde{\b}_\text{eff}} = -100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta5}
\caption{\it ${\tilde{\b}_\text{eff}} = 5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta100}
\caption{\it ${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta300}
\caption{\it ${\tilde{\b}_\text{eff}} = 300$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta10973-3}
\caption{\it ${\tilde{\b}_\text{eff}} = 10973.3$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1000_chi0.01/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it AdS, $\tilde{\a} = -1000$, $GN^2\chi^2 = 0.01$.}
\label{AdS_alpha-1000_chi0.01}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta-4-2}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.2$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta-4-15327}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.15327$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta-3-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -3.7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta-1-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -1.7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta50}
\caption{\it ${\tilde{\b}_\text{eff}} = 50$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta6089-09}
\caption{\it ${\tilde{\b}_\text{eff}} = 6089.09$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-100_chi0.001/beta100000}
\caption{\it ${\tilde{\b}_\text{eff}} = 100000$}
\end{subfigure}
\caption{\it AdS, $\tilde{\a} = -100$, $GN^2\chi^2 = 0.001$.}
\label{AdS_alpha-100_chi0.001}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-14-5653}
\caption{\it ${\tilde{\b}_\text{eff}} = -14.5653 $ (*)}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-8}
\caption{\it ${\tilde{\b}_\text{eff}} = -8$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-5-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -5.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-4-78}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.78$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta-1-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -1.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta2-2}
\caption{\it ${\tilde{\b}_\text{eff}} = 2.2$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi2pi/beta50}
\caption{\it ${\tilde{\b}_\text{eff}} = 50$}
\end{subfigure}

\caption{\it AdS, $\tilde{\a} = 0$, $GN^2\chi^2 = 2\pi$.}
\label{AdS_alpha0_chi2pi}
\end{figure}

\begin{figure}[h!]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta-100}
\caption{\it ${\tilde{\b}_\text{eff}} = -100$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta-9-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -9.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta-8}
\caption{\it ${\tilde{\b}_\text{eff}} = -8$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta-7-6}
\caption{\it ${\tilde{\b}_\text{eff}} = -7.6$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta-6-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -6.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta17-4347}
\caption{\it ${\tilde{\b}_\text{eff}}={\pi\b_\text{BF}\over N^2} = 17.4347$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_chi2Pi_am2/beta100}
\caption{\it ${\tilde{\b}_\text{eff}} = 100$}
\end{subfigure}
\caption{\it $\a = -2$, $GN^2\chi^2 = 2\pi$.}
\label{AdS_chi2pi_am2}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-12-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -12.5 $}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-12-3}
\caption{\it ${\tilde{\b}_\text{eff}} = -12.3$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-11-6854}
\caption{\it ${\tilde{\b}_\text{eff}} = -11.6854 (*)$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-10-5}
\caption{\it ${\tilde{\b}_\text{eff}} = -10.5$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-9}
\caption{\it ${\tilde{\b}_\text{eff}} = -9$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-6}
\caption{\it ${\tilde{\b}_\text{eff}} = -6$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta-4-7}
\caption{\it ${\tilde{\b}_\text{eff}} = -4.7$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha0_chi1000/beta50}
\caption{\it ${\tilde{\b}_\text{eff}} = 50$}
\end{subfigure}

\caption{\it AdS, $\tilde{\a} = 0$, $GN^2\chi^2 = 1000$.}
\label{AdS_alpha0_chi1000}
\end{figure}

\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta-30}
\caption{\it ${\tilde{\b}_\text{eff}} = -30$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta-14}
\caption{\it ${\tilde{\b}_\text{eff}} = -14 $}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta-12-6}
\caption{\it ${\tilde{\b}_\text{eff}} = -12.6$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta-10}
\caption{\it ${\tilde{\b}_\text{eff}} = -10$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta-1}
\caption{\it ${\tilde{\b}_\text{eff}} = -1$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta4-31457}
\caption{\it ${\tilde{\b}_\text{eff}} = 4.31457$ (*)}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\includegraphics[width=\textwidth]{Figures/AdS_alpha-1_chi1000/beta50}
\caption{\it ${\tilde{\b}_\text{eff}} = 50$}
\end{subfigure}

\caption{\it AdS, $\tilde{\a} = -1$, $GN^2\chi^2 = 1000$.}
\label{AdS_alpha-1_chi1000}
\end{figure}

\clearpage

% %
% %
% %
\addcontentsline{toc}{section}{References}

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\end{thebibliography}

\end{document}