[1]Cristina-Iulia<EMAIL_ADDRESS>
# Nanopublication-Based Semantic Publishing and Reviewing: A Field Study with
Formalization Papers

Cristina-Iulia Bucur  Vrije Universiteit Amsterdam, Amsterdam, The Netherlands
Tobias Kuhn  Vrije Universiteit Amsterdam, Amsterdam, The Netherlands  Davide
Ceolin  Centrum Wiskunde & Informatica, Amsterdam, The Netherlands  Jacco van
Ossenbruggen  Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

###### Abstract

With the rapidly increasing amount of scientific literature, it is getting
continuously more difficult for researchers in different disciplines to keep
up-to-date with the recent findings in their field of study. Processing
scientific articles in an automated fashion has been proposed as a solution to
this problem, but the accuracy of such processing remains very poor for
extraction tasks beyond the most basic ones (like locating and identifying
entities and simple classification based on predefined categories). Few
approaches have tried to change how we publish scientific results in the first
place, such as by making articles machine-interpretable by expressing them
with formal semantics from the start. In the work presented here, we set out
to demonstrate that we can formally publish high-level scientific claims in
formal logic, and publish the results in a special issue of an existing
journal. We use the concept and technology of nanopublications for this
endeavor, and represent not just the submissions and final papers in this RDF-
based format, but also the whole process in between, including reviews,
responses, and decisions. We do this by performing a field study with what we
call formalization papers, which contribute a novel formalization of a
previously published claim. We received 15 submissions from 18 authors, who
then went through the whole publication process leading to the publication of
their contributions in the special issue. Our evaluation shows the technical
and practical feasibility of our approach. The participating authors mostly
showed high levels of interest and confidence, and mostly experienced the
process as not very difficult, despite the technical nature of the current
user interfaces. We believe that these results indicate that it is possible to
publish scientific results from different fields with machine-interpretable
semantics from the start, which in turn opens countless possibilities to
radically improve in the future the effectiveness and efficiency of the
scientific endeavor as a whole.

##  1 Introduction

Considering the abundance of scientific articles that are published every day,
keeping up with the latest research is becoming a significant challenge for
researchers in many fields. This is at least partially due to the fact that we
are still holding on to an archaic paradigm of scientific publishing: the
canonical way to publish scientific results is by writing them up in long
English texts called articles, which are in the best case easy to read by
human experts but remain mostly inaccessible to automated approaches (except
on a very superficial level with text mining approaches). These articles then
undergo peer reviewing, which is typically done in a way that is secretive and
not standardized, with the effect that the reviewing process may lack
transparency and the valuable comments from the reviewers cannot be reused or
build upon. There have been studies on the effectiveness of peer-reviewing in
its current form (Smith,, 1988; Linkov et al.,, 2006; Kotturi et al.,, 2017)
that showed not only systematic biases among peer-reviewers, but also a lack
of transparency in the general peer-reviewing process as a whole (Smith,,
2010; Benda and Engels,, 2011; Lee et al.,, 2012). Making reviews open might
alleviate some of these concerns by ensuring higher-quality reviews, while at
the same time increasing the trust in the reviewing process and the quality of
the scientific publications themselves.

A range of approaches have been proposed to address some of these problems by
making scientific texts machine-readable, allowing for automatic summarising,
finding and retrieving information easier and even the ability to (partially)
reason on the scientific texts themselves. Text mining approaches work
reasonably well when it comes to simple entity extraction with techniques like
named-entity recognition to extract the main concepts from a text (e.g. (Al-
Moslmi et al.,, 2020; Yadav and Bethard,, 2018)), but accuracy dramatically
drops with more complicated tasks like relation extraction or identifying
links between entities (Etzioni et al.,, 2005; Xu et al.,, 2015; Zeng et al.,,
2014).

The vast majority of existing approaches of making scientific texts machine-
readable have one thing in common: they take the current paradigm of
scientific articles for granted and therefore take them as their starting
point to extract information. While it is important to try to process the vast
amount of existing scientific literature that has the form of long English
texts (and sometimes long texts in other languages), we should also think
about how we can improve the way how we publish scientific insights in the
first place. An important aspect of this is the vision of semantic publishing,
which we mean here in the sense of _genuine semantic publishing_ (Kuhn and
Dumontier,, 2017), where the machine-interpretable formal semantics cover the
main scientific claims the work is making. Nanopublications (Groth et al.,,
2010), which are small RDF-based semantic packages, have emerged as a powerful
concept and technology for enabling such genuine semantic publishing.

In previous research we have applied nanopublications to implement a semantic
and fine-grained model for reviewing (Bucur et al.,, 2019), and have extended
this to semantically represent the full structure of (classical) scientific
articles with their reviews and review responses as a single network of
nanopublications (Bucur et al.,, 2020). In order to get closer to our vision
of genuine semantic publishing, however, we need to represent not just the
structure but also the main content of these articles, most importantly their
main scientific claims. To that aim, we proposed in subsequent work the
_super-pattern_ , a semantic template to represent the meaning of scientific
claims in formal logic (Bucur et al.,, 2021).

Taking an example from our previous study as illustration of the super-
pattern, it has been stated in the scientific literature (Felix and Barrand,,
2002) that in particular kinds of cells in the rat brain (specifically,
endothelial cells) some sort of stress called transient oxidative stress
affects the expression of a protein called Pgp. The super-pattern consists of
five slots that would in this example be filled in as follows:

  * •

Context class: rat brain endothelial cell

  * •

Subject class: transient oxidative stress

  * •

Qualifier: generally

  * •

Relation: affects

  * •

Object class: Pgp expression

Informally, we can read this in the following way: whenever there is an
instance of transient oxidative stress in the context of an instance of a rat
brain endothelial cell, then generally (meaning in at least 90% of the cases),
that instance of stress has the relation of affecting an instance of Pgp
expression. Formally, it directly maps to this logic formula:

$P(\mbox{\leavevmode\nobreak\ }\exists z(\mbox{\small pgp-
expression}(z)\wedge\mbox{\small in-context}(z,x)\wedge\mbox{\small
affects}(y,z))\mbox{\leavevmode\nobreak\ }|\\\ \mbox{\leavevmode\nobreak\
}\mbox{\small transient-oxidative-stress}(y)\wedge\mbox{\small rat-brain-
endothelial-cell}(x)\wedge\mbox{\small in-
context}(y,x)\mbox{\leavevmode\nobreak\ })\geq 0.9$

This is stating in logic terms (in slightly non-standard notation using
conditional probability as a shorthand) that given a thing $y$ of type
_transient-oxidative-stress_ in the context of a thing $x$ of type _rat-brain-
endothelial-cell_ , the probability of there being a $z$ of type _pgp-
expression_ that is in the same context $x$ is at least 90%. We have shown
that this pattern can be applied to formalize most high-level claims found in
scientific literature across disciplines (Bucur et al.,, 2021).

In the work to be presented below, we combine all these elements of our
previous work — namely semantic representation of reviews, scientific works as
a networks of nanopublications, and representing the main claims with the
super-pattern — in order to implement genuine semantic publishing and putting
it to the test in a field study. For practical reasons, we did not require the
scientific claims in this field study to be novel ones, but they were selected
from existing publications. This field study led to the publication of a
special issue in an established journal (Data Science) at an established
publisher (IOS Press). This special issue consists of what we call
_formalization papers_ , which are nanopublication-based semantic publications
whose novelty lies in the formalization of a previously published scientific
claim.

In this research we therefore aim to answer the following research question:

  * •

Are nanopublications and the super-pattern appropriate concepts to enable a
new paradigm of scientific communication where authors publish their
scientific findings with formal semantics?

The rest of this article is structured as follows. In Section 2 we describe
the current state of the art in the field of scientific publishing with regard
to scientific knowledge representation, semantic publishing and semantic
articles and also alternative proposed machine-readable approaches like
nanopublications. In Section 3 we describe our approach with regard to a new
way of publishing, starting from a formal way of representing the content of
scientific claims and ending with the representation of the publication
process itself in what we call “formalization papers”. We then report and
discuss the results of the field study we performed using formalization papers
in Section 4. Future work and conclusion of the present research are outlined
in Section 5.

##  2 Background

We provide here the background on scientific knowledge representation,
scientific publishing, and nanopublications in particular.

###  2.1 Scientific knowledge representation

Novel proposals for the current “Disruption Era” (Rahardja et al.,, 2019)
include scientific publication management models that connect abstract
knowledge with actual world problems in the constantly growing body of
scientific knowledge (Chi et al.,, 2018), and the use of decentralized
publication systems for open science using, for example, existing technologies
like Blockchain and IPFS (Tenorio-Fornés et al.,, 2019).

With respect to the growth of scientific literature, there is a trend towards
multidisciplinary and interdisciplinary research, together with an increased
volume of publications every year (Wong,, 2019). A range of methods have been
proposed to make scientific articles more machine-readable: from structuring
scientific works as Research Objects (RO) (Bechhofer et al.,, 2013; Belhajjame
et al.,, 2015) to using facets in order to uncover the main methods, data,
code and other objects that are used in scientific articles (Peroni et al.,,
2013; McGregor,, 2008). Most approaches, however, have focused on automated
content extraction from scientific articles as they are currently available.
Recent machine learning techniques, for example, can after training with large
sets of scientific articles extract the main concepts and structure of
scientific articles (Xu et al.,, 2015; Zeng et al.,, 2014). While the results
can be very valuable there are also clear limitations, with the resulting data
needing almost always manual curation to achieve decent quality (Garcia-Castro
et al.,, 2013; Coulet et al.,, 2011; Sernadela et al.,, 2015).

A significant number of vocabularies and ontologies in many various domains
have been developed, which are now ready to be used for scientific knowledge
representation. However, they remain often difficult to find, access and
understand due to the lack of documentation, versioning problems, and
unresolvable URIs, among other things (Garijo and Poveda-Villalón,, 2020;
Halpin et al.,, 2010; Hitzler and van Harmelen,, 2010; Jain et al.,, 2010). A
considerable amount of attention has also been given to the datasets
accompanying scientific articles. The Data Set Knowledge Graph (DSKG), for
example, covers datasets from over 600k scientific publications (Färber and
Lamprecht,, 2021). An important development in this respect is the strong
momentum behind the FAIR initiative to make research data Findable,
Accessible, Interoperable, and Reusable (Wilkinson et al.,, 2016). A large
amount of research is ongoing on how these FAIR principles can be put into
practices (e.g. (Garijo and Poveda-Villalón,, 2020)). Many other aspects of
scientific communication have been approached with more formal
representations, such as declaring authorship contributions with the
Contributor Roles Taxonomy (McNutt et al.,, 2018) to mention just one of them.

Semantic technologies have been used extensively in the Life Sciences, e.g.
for the representation and discovery of concepts, their relationships and
associated supporting evidence in order to integrate distributed repositories
(Hannestad et al.,, 2021). A variety of controlled vocabularies exist in these
fields that can serve as the foundation to represent scientific knowledge in a
structured way in order to semantically capture the context of scientific
findings (Chibucos et al.,, 2014; Slater and Song,, 2012; Madan et al.,,
2019). The BEL language (Slater,, 2014) is one of the few attempts to
represent the high-level scientific claims themselves, with coverage for
specific kinds of biological relations. Also many other domains besides the
Life Sciences have adopted the principles and technologies of Linked Data and
the Semantic Web, for example to build interlinked, heterogeneous, and
semantically rich datasets in Cultural Heritage (Hyvönen,, 2012) and to find,
address, and sometimes even solve research problems in Digital Humanities in
interactive ways (Hyvönen,, 2020).

###  2.2 Semantic publishing and semantic papers

Semantic publishing applies semantic technology to scientific publishing, and
comes in many forms and does not always align with what we have introduced
above as genuine semantic publishing (Kuhn and Dumontier,, 2017). Under this
umbrella of semantic publishing, there are approaches that generate
semantically-enriched data models from digital publications for the
integration, sharing, management and data comparison between publications
(Perez-Arriaga,, 2018), study the semantic annotation and enhancement of
scholarly articles (Shotton,, 2009), provide dynamic visualizations in
semantically enhanced papers (Senderov and Penev,, 2016), assess the
versioning aspect of semantic publishing (Papakonstantinou et al.,, 2018),
create a global-scale platform with a dataset metadata for automated
ingestion, discover, and linkage (Jacob et al.,, 2017), and propose semantic
and web-friendly HTML-based alternatives to the currently PDF-focussed
scientific writing process (Peroni et al.,, 2016). Semantic enhancements of
scientific articles can be used for semantic interlinking, interactive
figures, re-orderable references and even summary creation (Shotton et al.,,
2009), and workflows to convert regular scientific articles into Linked Open
Data have also been investigated (Sateli and Witte,, 2016). Other approaches
like the compositional and iterative semantic enhancement (CISE) advocate for
a process of automatic semantic enhancement with semantic annotations
(Peroni,, 2017). A key role in most of these approaches is played by the
variety of existing ontologies covering many different aspects of scientific
publishing, most importantly the Semantic Publishing and Referencing (SPAR)
Ontologies (Peroni,, 2014).

Further note-worthy approaches include the work to semantically represent the
setup and results of scientific studies, which then allows for running meta-
analyses in a semi-automated way, better research replication, and automated
hypothesis generation (Tiddi et al.,, 2020), and the development of the Open
Research Knowledge Graph (Jaradeh et al.,, 2019). The latter is an initiative
that aims to make research articles machine-readable by expressing their main
scientific entities as a semantically interconnected knowledge graph. This
graph is populated by methods such as extracting scientific concepts from the
abstracts of scientific articles with the help of annotators (Brack et al.,,
2020).

###  2.3 Nanopublications

Nanopublications (Groth et al.,, 2010) are a specific concept and technology
that deserves special attention here. They have been proposed to express
scientific (and other kind of) knowledge in Linked Data as small independent
publication packages. They allow for rich provenance and metadata and are
structured as follows: the assertion part contains the main content of the
nanopublication, such as a scientific claim, expressed as RDF triples. The
provenance part of a nanopublication describes how the assertion came about,
e.g. by linking to the scientific methods used to arrive at the finding. The
publication information part, finally, contains metadata about the
nanopublication as a whole, such as by whom and when it was created.
Nanopublications can be used for scientific findings, but also for
representing the other elements of the scientific workflow, such as reviews
and method descriptions, and more generally any kind of small coherent set of
RDF triples (Kuhn et al.,, 2013). It has been shown how nanopublications can
be made reliable and immutable by identifying them with cryptographic Trusty
URIs (Kuhn and Dumontier,, 2015), and how this allows for a decentralized
network of services and template-based user interfaces such as Nanobench (Kuhn
et al.,, 2021).

##  3 Methodology

In this section we describe the approach and methods we followed to
investigate whether nanopublications and the super-pattern are suitable to
achieve genuine semantic publishing.

###  3.1 Approach

In our approach, we committed to a number of features. First, we wanted the
final contributions to be published as “real” papers in a real established
journal. They should be fully semantically represented (in RDF) but also have
classical views that makes them look like other papers. Like that, they should
also seamlessly integrate with the existing bibliometric system and it should
be straightforward to cite them in the classical way.

Second, we decided to fully focus on arguably the most interesting element of
scientific articles, which happens to also be one of the most challenging to
formally represent: the main scientific claims the article is making.
Scientific articles have a large number of other interesting pieces of
information, e.g. information about the used methods among many other things,
but for the purpose of the study to be presented, we focus only on the main
claims.

Third, in order to retain the flexibility and power of nanopublications, we
decided to refrain from providing a custom-built and optimized user interface
that hides the complexity and limits the flexibility. By using generic
template-based nanopublication tools and by customizing them solely by
providing the templates, we hoped to get a better understanding of how the
nanopublication technology works for such kinds of content and workflows in
general, and not just for our specific case. On the other hand, this also
means that we were looking for a bit more technically minded authors who can
handle interfaces that do not come with all the comfort of polished specific
applications.

Fourth, we wanted to test a system that _could_ be used to publish novel
claims, but decided for practical reasons to focus on formalizing claims from
previously published articles. Our approach is therefore based on what we call
_formalization papers_ that contribute novel formalizations of existing
claims.

Finally, we wanted to cover not just these main claims, but the whole
publishing workflow that involves the initial submission of contributions,
their reviewing, the responses to the reviews, the updated versions, and the
final decision, and represent these as independent but interlinked
nanopublications.

###  3.2 Formalization Papers

Our approach builds upon our new concept of formalization papers. A
formalization paper contributes a semantic formalization of one of the main
claims of an already published scientific article. Its novelty therefore lies
solely in the formalization of a claim, not the claim itself. The authors of
such formalization papers consequently take credit for the way how the
formalization is done, but not for the original claim (unless that claim
happens to come from the same authors).

Figure 1: Formalization paper template from Nanobench as used by the
participants of our study.

The content of a formalization paper is fully expressed in RDF in the form of
nanopublications. Such a formalization paper can be shown in other formats to
users, e.g. in HTML or PDF, but these are just views of the same underlying
RDF content. Our formalization papers consist of nanopublications in which the
assertion contains the formalization of the scientific claim using the super-
pattern (Bucur et al.,, 2021), the provenance points to the original paper of
the claim, and the publication information attributes the author of the
formalization. Figure 1 shows an example of such a nanopublication in the
interface the participants of our study used to create them. The instantiated
super-pattern in the assertion part refers to a context class, a subject
class, a qualifier, a relation type, and an object class according to the
super-pattern
ontology111https://larahack.github.io/linkflows˙superpattern/doc/sp/index-
en.html. In the process of coming up with such a formalization, one often
realizes that for some of the class slots of the super-pattern (i.e. context,
subject, and object class) the class that should be filled in to arrive at a
correct formalization is not directly defined in any existing vocabulary or
ontology and as such, this class might need to be minted as well. The
provenance part of the nanopublication describes the “formalization activity”
that was conducted in order arrive at this formalization from what is written
in the source publication. The precise phrase from that source publication
that was used can be quoted too.

###  3.3 Tools

In order to publish formalization papers, class definitions, and all the other
kinds of nanopublications (submissions, reviews, responses to reviews, and
decisions), we use Nanobench (Kuhn et al.,, 2021)222https://github.com/peta-
pico/nanobench. Figure 1 introduced above shows a screenshot of the publishing
page of Nanobench. Publishing in Nanobench is based on templates, which are
themselves expressed in nanopublications. The form shown in the screenshot is
automatically generated based on the information found in several template
nanopublications that we created and published for that purpose. All the
application-specific behavior is therefore semantically represented in the
templates, and Nanobench can flexibly be used for any other kind of data and
workflow.

Figure 2: The Tapas interface listing submitted formalizations as the results
of SPARQL queries over the nanopublication service network.

The second tool that we are using, Tapas (Lisena et al.,,
2019)333https://github.com/peta-pico/tapas, is equally generic. It is a simple
user interface component built on top of grlc (Meroño-Peñuela and Hoekstra,,
2016) that allows to run template-based SPARQL queries on RDF triple stores.
In our case, we run it on SPARQL endpoints provided by the nanopublication
service network (Kuhn et al.,, 2021). We use Tapas to show aggregations and
overviews of submissions and reviews. Figure 2 shows a screenshot of the main
submission overview. Tapas by itself is read-only, but we connect to the
Nanobench tool with links that lead to partially filled-in forms (e.g. “click
here to add review” in the screenshot).

###  3.4 Field study design

In order to test our approach, we devised a field study where interested
authors could submit formalization papers, which upon acceptance are to be
published as a special issue in the journal Data
Science444https://datasciencehub.net/ by IOS Press. The goal of this was to
demonstrate for the first time that scientific articles can be formalized and
therefore machine-interpretable including the main scientific claims. As a
secondary goal, we wanted to find out whether nanopublications are a good
technology for that, and whether it is feasible to represent also the entire
submission and reviewing process within the same framework.

Because the user interfaces we have at our disposal are still quite rough and
technical, we restricted the set of possible authors and sent the call for
papers on a by-invitation basis to selected groups of researchers who have
previously worked or had experience with technologies like RDF and semantics.
We expect to be able to build more accessible user interfaces in the future
that can show the inherent complexity in a way that does not require technical
skills, but how this can be achieved is out of scope for this work.

Participants to our field study, thus the authors of formalization papers,
formalized their own previously published claim, or a claim from a paper
published by others. In the latter case, the formalization paper authors take
credit for the formalization of the claim but not for the claim itself. All
submissions to this special issue were peer-reviewed (also as
nanopublications) using our previously proposed reviewing ontology (Bucur et
al.,, 2019). Upon acceptance, these formalization papers are to be published
in a journal at IOS Press, thereby giving them the same bibliometric status as
other scientific articles, which leads to regular indexing in scientific
article databases, counting of citations, and so on.

Figure 3: Timeline publication for the special issue with formalization papers
at the Data Science journal.

The whole timeline of the field study that encompasses the special issue can
be seen in Figure 3. The authors received close guidance on how to represent a
claim of their choosing in RDF using the super-pattern and nanopublications,
and on the various stages of the publication process. Authors took part in
several information sessions and discussion meetings and were provided at each
step with helper materials, videos, and even direct assistance if needed. In
total, 24 such individual sessions were organized from May to December 2021.

In order to define a formalization, sometimes some of the class slots (i.e.
context, subject, and object slots) of the super-pattern should be filled in
with classes that are not yet defined in any existing vocabulary or ontology.
In this case the authors first had to define these themselves, and they could
do that also with the Nanobench tool loading a template for class definition.
(Alternatively, they could also mint a new class identifier by other means,
such as creating it on Wikidata.) The assertion of a nanopublication defining
a new class may look for example as follows (link to full nanopublication):

      sub:STX1B-mutation a owl:Class ;
        rdfs:subClassOf wd:Q42918 ;
        rdfs:label "STX1B mutation" ;
        skos:definition "mutation in STX1B" ;
        skos:relatedMatch wd:Q18048867 .

Here, “mutation” from Wikidata (Q42918) is declared as super-class of the
newly minted class “STX1B mutation”, and “STX1B” (Q18048867) is linked as a
related class.

Then the authors can publish their formalization in the form of a
nanopublication using Nanobench (see Figure 1), and afterwards they needed to
submit it to the special issue using another Nanobench template, leading to an
assertion like (link to full nanopublication):

      <http://purl.org/np/RAGo62Hb_Bx1klF4pn1q1Ty40860e3A7Sz4hr2vojZ2wA>
        pso:withStatus pso:submitted ;
        frbr:partOf fpsi:DataScienceSpecialIssue .

All submitted formalizations were subsequently reviewed. All authors were
encouraged to review other submissions, and these reviews were semantic, open,
and non-anonymous. These reviews were again done in nanopublications with the
Nanobench tool. Such an example of a nanopublication assertion that contains a
review modeled using the reviewing ontology can be seen below (link to full
nanopublication):

      sub:comment a lfr:ReviewComment , lfr:ContentComment , lfr:NeutralComment ,
          lfr:SuggestionComment ;
        lfr:hasCommentText "Maybe the use of a causal relation like \"contributes to\"
                             can also be used here." ;
        lfr:hasImpact "1" ;
        lfr:refersTo <http://purl.org/np/RAGo62Hb_Bx1klF4pn1q1Ty40860e3A7Sz4hr2vojZ2wA> ;
        lfr:refersToMentioningOf sp:hasRelation .

In such a structured review (see more details in our previous research (Bucur
et al.,, 2020)), it is possible to specify various aspects that the review
addresses including the aspect it comments on (syntax, style or content), the
positivity/negativity of the review, the impact and the action that needs to
be taken by the authors as the reviews see it (whether it is compulsory to be
addressed, a suggestion or no action needs to be taken by the authors) and the
importance of the point made by the review for the overall quality of the
formalization. In the above example, the review comment makes a neutral point
about the content of the given formalization with an importance of 1 out of 5,
and is marked as a suggestion for the authors. The specific part of the
formalization that this review targets is the _sp:hasRelation_ field, as
indicated by the _refersToMentioningOf_ relation.

Subsequently, authors of the submissions could respond to the received review
comments, again in nanopublications, and update their submissions based on
these review comments. This is an example of a response to a review comment
(link to full nanopublication):

      sub:comment a , lfr:ResponseComment lfr:DisagreementComment ,
          lfr:PointNotAddressedComment ;
        lfr:hasCommentText "I don’t think the original publication shows a causal
                            relationship. It seems to me only a correlation is proven." ;
        lfr:isResponseTo
          <http://purl.org/np/RAio--7IbPa3_ZSG3GspUsXeWP2ZwMIzy4Kzos0yZ7NIw> ;
        lfr:refersTo <http://purl.org/np/RAeRSya2qIYymsBxiqOZP_oaQpHXUVXiydKvPCFM-7DDQ> .

This response registers the agreement with the point made by the reviewer
(whether the author agrees totally, partially or not at all) and if that point
was addressed, partially addressed or not addressed at all by the author.
Moreover, a link to the respective review is given using the _isResponseTo_
relation, while the updated version of the formalization is indicated using
the _refersTo_ relation. In our example, we see that the author does not agree
with the point made by the reviewer and hence did not address the point raised
by him, and also give a textual motivation on why this is the case.

Figure 4: A nanopublication containing a formalization paper.

Finally, the authors updated their formalizations with the same template as
depicted in Figure 1. The full final formalization nanopublication of the same
example is shown in Figure 4. For all updated submissions then a decision was
made by us as the special issue editors about their acceptance. This decision
was also represented as a nanopublication that looked as follows (link to full
nanopublication):

      <http://purl.org/np/RAeRSya2qIYymsBxiqOZP_oaQpHXUVXiydKvPCFM-7DDQ>
        dct:description "All review comments were addressed and the formalization looks
                         good." ;
        pso:withStatus pso:accepted-for-publication ;
        frbr:partOf fpsi:DataScienceSpecialIssue .

All formalizations reached a satisfactory level of quality, as indicated by
the reviews and the authors’ responses, and we therefore accepted all 15
submissions for publication.

Figure 5: The “classical view” of a formalization paper, as it will appear on
the publisher’s website.

In order to show the accepted papers in the special issue as if they were
classical papers, to integrate them in the publisher’s content management
system, and to make them connect to the existing bibliometric system, we semi-
automatically created “classical views” in the form of HTML and PDF versions
of the nanopublications, as can be seen in Figure 5.

###  3.5 User Feedback

In order to evaluate the general idea of formalization papers, all
participants to the field study were asked to give us their opinion and report
on their experiences about the involved processes and concepts. This
evaluation was performed by means of a structured questionnaire consisting of
four main parts, each one evaluating different aspects of the workflow.

In the first part, we are interested in assessing the difficulty of
conceptually understanding the formalization paper idea and the super-pattern,
and of performing the formalization tasks. In part two, we focus on the
difficulty of the technical aspects in the various submission, reviewing and
revision stages. Part three addresses some more general aspects about the
authors’ experience and preferences. Authors were asked about their confidence
in the formalization they published and about their interest of publishing
such formalizations along their scientific publications in the future. We also
asked them how important they think it is that all these steps are performed
by the authors themselves (as they did). Moreover, in this part, authors could
give us their opinion with regard to the importance of having a “classical
view” along with the nanopublication representation of their formalization
paper. The fourth and final part of the questionnaire asked for the technical
background of the authors. At the very end, the respondents could give further
free-text feedback. The full questionnaire is available in our supplemental
material555https://github.com/LaraHack/formalization˙papers˙supplemental/tree/main/questionnaire.

##  4 Results

In this section we present the formalizations that resulted from our field
study. We present a descriptive analysis of the generated data and analyze it
also with the help of a network visualization. Finally, we report on the
results from the user feedback questionnaire.

All the nanopublications that were created for all the submissions,
formalization paper versions, the review comments, the responses to the review
comments and the newly minted classes used in the formalizations together with
the decisions are accessible
online666https://github.com/LaraHack/fpsi˙analytics/tree/main/nanopubs, while
the nanopublication index containing all these nanopublications has also been
published777Nanopublication index:
http://purl.org/np/RAkLJW7vIsnKKJDf1iswdgtFPQSo3lEG˙z8DhHfD7dofE. Also, the
final submissions for the special issue with formalization papers at the Data
Science journal 888The link will be added in late March 2022, when the special
issue is due for publication. can be found
online999https://github.com/LaraHack/formalization˙papers˙supplemental/tree/main/accepted˙submissions.

###  4.1 Analysis of Formalizations

In total, we had an initial number of 20 people that replied to our call for
papers101010https://github.com/LaraHack/formalization˙papers˙supplemental/tree/main/call˙for˙papers
from 12 different institutions from the United States of America, Germany,
Luxembourg, Bulgaria, and The Netherlands from fields like biomedicine,
bioinformatics, health sciences, ecology, data science, and computer science.
After an initial information session, out of the 20 authors that responded to
the call for papers, 18 decided to continue their participation. All these 18
authors that responded to the call for formalization papers managed in the end
to publish (upon acceptance) their articles in a special issue at the Data
Science journal.

We had a total of 15 formalization paper submissions, 13 with individual
authors and 2 with joint authorship. Out of the total of 18 authors, two of
these have both an individual submission and a joint-authorship one. The
super-pattern instantiations of the final accepted formalization paper
submissions can be seen in Table 1. Here, the classes used to instantiate the
super-patterns that comprise the formalizations are given for each submission:
the context, subject and object classes for each submission are listed,
together with the qualifier and relations selected from the SuperPattern
ontology (Bucur et al.,, 2021). Each instantiation of the super-pattern can be
interpreted as follows: “Every thing of type [SUBJECT] that is in the context
of a thing of type [CONTEXT] [QUALIFIER] has a relation of type [RELATION] to
a thing of type [OBJECT] that is in the same context.”.

Looking at Table 1, we see that the super-pattern instances exhibit quite a
broad variety of scientific fields (bioinformatics, biomedicine, pharmacology,
data science, computer science) mostly linked to the life sciences. 7 out of
the 15 submissions contain a formalization in which authors extracted a
scientific claim from their own previously published article (submission
number marked with $\diamond$). Additionally, out of the total 44 classes used
in the formalizations, 22 new classes were minted using Nanobench (marked with
*), while 4 were newly minted Wikidata classes (marked with **). 13 already-
existing classes were reused from Wikidata (their Wikidata identifier is
specified next to the class name) and 4 classes were referenced from other
ontologies.

|  CONTEXT |  SUBJECT |  QUALIFIER |  RELATION |  OBJECT
---|---|---|---|---|---
|  (“in the context of all …”) |  (“things of type …”) |  |  |  (“to things of type…”)
1 |  early human adipogenesis* |  regulatory element within the first intron of FTO* |  generally |  affects |  expression of genes IRX3 and IRX5*
2 |  human motor neuron (Q101404862) |  TAR DNA binding protein (Q21133247) |  can generally |  contributes to |  transcription of stmn2*
3 $\diamond$ |  dejellied fertilizable stage VI Xenopus laevis oocyte** |  strong static magnetic field** |  generally |  affects |  cell cortex (Q5058180)
4 $\diamond$ |  (no context class) |  genes associated with CAKUT** |  sometimes |  is same as |  targets of vitamin A**
5 $\diamond$ |  patient undergoing PCI* |  pharmacogenomics guided clopidogrel therapy* |  generally |  enables |  cost-effective treatment*
6 |  human (Q5) |  smoothened signaling pathway |  mostly |  affects |  astrocyte development
7 $\diamond$ |  biodiversity data (Q28946370) |  license with non-commercial clause* |  generally |  inhibits |  data reuse (Q58023280)
8 $\diamond$ |  release of OpenBiodiv knowledge graph* |  triple in OpenBiodiv knowledge graph* |  generally |  is same as |  semantic triple extracted from biodiversity literature*
9 |  UNC13A (Q18036664) |  TAR DNA binding protein (Q21133247) |  generally |  inhibits |  inclusion of cryptic exon
10 $\diamond$ |  data set (Q1172284) |  adherence to the FAIR guiding principles* |  can generally |  enables |  automated discovery*
11 |  human (Q5) |  NGLY1 deficiency |  always |  is caused by |  dysfunction of ERAD pathway*
12 |  social group (Q874405) |  relative neocortex size* |  never |  affects |  social group size*
13 |  ecm bound cancer cell* |  glycocayx bulk* |  generally |  increases |  integrin clustering*
14 |  human (Q5) |  STX1B mutation* |  frequently |  co-occurs with |  epilepsy (Q41571)
15 $\diamond$ |  digital humanities research* |  usage of Linked Data Scopes* |  can generally |  contributes to |  transparency (Q535347)
Table 1: Instantiated super-patterns accepted for publication in formalization
papers in the Data Science special issue. Submissions marked with $\diamond$
are formalizations in which authors extracted a scientific claim from their
own previously published article; classes minted using Nanobench are marked
with *, while newly minted Wikidata classes are marked with **.

###  4.2 Analysis of Nanopublications

In Table 2 shows the statistics about the nanopublications created during our
field
study111111https://github.com/LaraHack/formalization˙papers˙supplemental/tree/main/nanopubs.
It shows a total of 15 submissions with their 15 corresponding super-pattern
definitions. There are 25 updated super-patterns, indicating that some of the
submissions were updated more than once. 34 new classes were minted in
nanopublications as class definitions, which were subsequently used in the
formalizations.

With regard to the reviews received and the author responses, class
definitions received an average number of around 3 reviews per class, while
the super-pattern definitions had almost 8 review comments on average. In
terms of the responses given to these reviews, the average responses to class
definitions was a little over 2, while the average number of responses to the
review comments for the super-pattern definitions was about 6.7.

icon | type | per submission | total
---|---|---|---
| submissions | 1.00 | 15
| super-pattern definitions | 1.00 | 15
| class definitions | 2.27 | 34
| reviews of super-patterns | 7.93 | 119
| reviews of class definitions | 3.07 | 46
| responses to super-pattern reviews | 6.67 | 100
| responses to class definition reviews | 2.27 | 34
| updated super-pattern definitions | 1.67 | 25
| decisions | 1.00 | 15
Table 2: Nanopublications created during the field study of the special issue
with formalization papers at the data Science journal.

In Figure 6 we can see this graphical representation of all the special issue
nanopublications, where each node represents such a nanopublication and the
arrows between the nodes show how the nanopublications are linked semantically
with each other. The legend for the node types indicated by color and letter
code can be found in Table 2.

For every formalization paper, we see a first formalization (F) together with
a submission nanopublication (S). Later updated versions (U) of formalizations
also link back to the initial formalization. The initial submissions received
review comments (R), to which authors then answered with response
nanopublications (A). Additionally, some of the formalization papers used
newly minted classes (C), which then also received review comments and
responses. The final decision (D) points to the finally accepted updated
formalization. The edges (i.e. arrows) of the graph indicate when a
nanopublication is referring to another one by using its identifier in the
assertion. The edges shown in red are _superseding_ relations, pointing from a
new version of a nanopublication to its previous version. This is how
nanopublications, being immutable, are dealing with representing new versions.

Figure 6: Graphical representation of all the submissions to the Data Science
special issue with formalization papers. A click-able version with links to
the nanopublications can be found online:
https://raw.githubusercontent.com/LaraHack/fpsi˙analytics/main/np-graph.svg

###  4.3 User Feedback Analysis

The 18 authors and co-authors of the formalization papers were asked to fill
in the user feedback questionnaire. It was important for this questionnaire to
be fully and reliably anonymous, as the authors needed to be able to give
their honest opinions. This meant that we had to send reminders without
knowing who already filled it in. After several rounds of reminders, we ended
up getting 19 responses, meaning that at least one of the authors submitted
two responses. Due to the anonymous nature of this questionnaire, it was not
possible find out which responses were affected, and we have therefore to deal
with such a dataset of slightly imperfect representation.

How difficult or easy was it for you to CONCEPTUALLY understand …

[max=5,min=1,step=1,width=5cm,scale=0.8] [label=… what a formalization paper
is?]4.32 [label=… the purpose of the super-pattern?]3.89 [label=… the role of
the context class?]3.58 [label=… the role of the subject class?]3.89 [label=…
the super-pattern qualifiers (”generally”, etc.)?]3.68 [label=… the super-
pattern relations (”causes”, etc.)?]3.89 [label=… the role of the object
class?]3.95 [label=… the overall interpretation of the super-pattern?]3.74

How difficult or easy was it …

[max=5,min=1,step=1,width=5cm,scale=0.8] [label=… to find an article with a
claim to formalize?]3.74 [label=… to understand what the claim exactly
meant?]3.74 [label=… to decide on the context class?]2.89 [label=… to decide
on the subject class?]2.79 [label=… to decide on the super-pattern
qualifier?]3.16 [label=… to decide on the super-pattern relation?]3.11
[label=… to decide on the object class?]3.16 [label=… to conceptually
represent the claim with the super-pattern overall?]2.79 $\leftarrow$
difficult / easy $\rightarrow$

Figure 7: Questionnaire Part 1: Conceptual Aspects.

How difficult or easy was it for you …

[max=5,min=1,step=1,width=5cm,scale=0.8] [label=… to setup Nanobench?]3.84
[label=… to use Nanobench?]3.32 [label=… to use Tapas?]2.76

How difficult or easy was it for you with the given tools (Nanobench and
Tapas) …

[max=5,min=1,step=1,width=5cm,scale=0.8]
[label=… at the SUBMISSION stage…] [label=… to define new classes?]3.11
[label=… to select the right qualifier, relation and classes for the
formalization?]3.26 [label=… to fill in the provenance part (in red)?]3.37
[label=… to publish your formalization?]3.95 [label=… to submit your
formalization to the special issue?]3.63
[label=… at the REVIEWING stage…] [label=… to publish review comments for
others?]3.06 [label=… to view received review comments?]3.21
[label=… at the FINAL REVISION stage…] [label=… to update your classes in
response to received review comments?]2.95 [label=… to update your
formalization in response to received review comments?]2.95 [label=… to
respond to review comments?]3.16 $\leftarrow$ difficult / easy $\rightarrow$

Figure 8: Questionnaire Part 2: Technical Aspects.

General aspects:

[max=5,min=1,step=1,width=5cm,scale=0.8] [label=Confidence as author in the
quality of the formalization]4.00 [label=Importance that authors create final
formalization themselves]3.68 [label=Interest in publishing such
formalizations together with future articles]4.05 [label=Importance of
“classical view” for website visitors]4.68 [label=Importance of
“nanopublication view” for website visitors]3.32 $\leftarrow$ not at all /
very $\rightarrow$

How would you rate your knowledge with respect to the following topics?

[max=5,min=1,step=1,width=5cm,scale=0.8] [label=Knowledge representation]3.89
[label=Knowledge graphs/Linked Data]3.95 [label=Ontologies/vocabularies]4.00
[label=Nanopublications]3.47 [label=Formal logic]3.05 [label=Programming
languages]3.47 $\leftarrow$ none / expert $\rightarrow$

Figure 9: Questionnaire Parts 3 and 4.

In Figure 7 we see the results for the first part of the questionnaire.
Authors expressed that it was rather easy to understand what a formalization
paper is (with a score of 4.32 out of 5). The elements of the super-pattern
were found a bit harder to understand but still quite easy, with scores above
3.50. Finding an article from which to select a claim to formalize and to
understand what the chosen claim really meant was also deemed easy, with
scores of 3.74. The actual instantiation of the super-pattern with all its
fields given the chosen claim was considered a little more difficult, with
scores around 3.0, indicating medium difficulty roughly in the middle of _very
difficult_ and _very easy_. These results seem to suggest that the authors
were able to understand the main formalization papers idea together with the
super-pattern that comprises it, but when it came to the actual instantiation
of the super-pattern (especially concerning the context and subject class),
this was considered a little more difficult, but still on average far from
very difficult.

In Figure 8, we see the authors’ responses with respect to technical
difficulty. In terms of the tools used, we see that setting up and using
Nanobench was considered easy enough (with a score of 3.30), while the Tapas
interface seems a little harder to use (with a score of 2.76). The different
tasks in the different stages all seemed to be between medium and easy on
average, with the exception of the tasks to provide responses to reviews,
which scored slightly below 3.0. The response nanopublications are indeed
among the most complex ones, as they refer not only to the affected review but
also to the updated formalization. Overall, while these results show room for
improvement, they still seem favorable given that we were building upon
generic and powerful tools without specific user interface design or
polishing.

Figure 9 summarizes the assessment of more general aspects of formalization
papers and also contains information about the authors’ background. We see
that authors have a high confidence in the quality of their formalization,
with an average score of 4.0, and that they are interested in the future
publication of such a formalization along their scientific publications, with
a score of 4.05. The respondents very clearly stated that the classical view
of formalization papers is important for website visitors, with a score of
4.68. Exposing also the “naked” nanopublications to the website visitors with
a nanopublication view was found to be much less important (3.32).

The authors indicated that they have, on average, a high level of knowledge on
the topics of knowledge representation, knowledge graphs, Linked Data, and
ontologies/vocabularies, with scores from 3.89 to 4.00. Their background in
nanopublications, formal logic, and programming languages was significantly
lower, on average, but still relatively high, with scores between 3.05 and
3.47.

10 out of the 18 authors used the free text feedback of the questionnaire. 8
of these 10 respondents expressed their excitement about the field study and
found the formalization paper concept and the whole publication process
interesting and useful. However, half of these respondents also mentioned that
the overall process proved to be a little more difficult than they expected,
due to the tools used maybe being too technical. One author also pointed out
that multiple formalizations can be written for the same claim by choosing the
context, subject and object classes differently and expressed the worry that
this would decrease the interoperatibility or utility of formalizations
especially when aggregating or mining them. This is a reasonable point to
make, but due to the fully formal semantics, syntactic differences are in
principle not hindering this kind of interoperability. Overall, the super-
pattern, the formalization paper concept, and the nanopublication-based
publication workflow seem to have been well-accepted and understood by the
participants, and many of them showed an enthusiastic reaction.

##  5 Discussion and Conclusion

The publication of the special issue with formalization papers at the Data
Science journal shows not only that nanopublications and the super-pattern can
be used to implement the basic steps and entities of a journal workflow, but
also that authors of such formalization papers can be taught to use these in
order to publish in a novel journal publication workflow as the publication of
the special issue demonstrates. Our results show that the super-pattern can be
well understood conceptually and despite the fact that from a practical
standpoint applying it seems to be more difficult, its application remains
perfectly feasible. Furthermore, we saw in our field study that even if the
current general-purpose tools can be considered a viable solution, these are
not necessarily easy to use, but they still remain a good tool for the purpose
of publishing formalization papers. Moreover, considering the formalization
papers, authors seem confident with regard to the quality of their
publications and seem interested in publishing such formalizations in the
future.

In future work, we plan to take the next logical step by publishing novel
claims in this way form the start, and not depend on claims from already-
published papers. These contributions will then also have to be accompanied by
statements about the methods, equipment, and all other relevant scientific
concepts, and can include not just the high-level claim but more lower-level
ones, possibly all the way down to the raw data. This representation would
then ideally cover the entire scientific workflow, starting from a motivation,
leading to the design and execution of a study, and ending in new scientific
insights. Such fully formalized scientific contributions can be seen as a
major step — even a breakthrough — for the Semantic Web and Open Science
movements and will bring us closer to a world where machines can interpret
scientific knowledge and help us organize and understand it in a reliable and
transparent manner.

####

Acknowledgements. This research was partly funded by IOS Press and the
Netherlands Institute for Sound and Vision. The authors would like to thank
Stephanie Delbeque, Maarten Fröhlich and Johan Oomen for providing their
insight and expertise.

## References

  * Al-Moslmi et al., (2020) Al-Moslmi, T., na, M. G. O., Opdahl, A. L., and Veres, C. (2020).  Named entity extraction for knowledge graphs: A literature overview.  IEEE Access, 8:32862–32881.
  * Bechhofer et al., (2013) Bechhofer, S., Buchan, I., De Roure, D., Missier, P., Ainsworth, J., Bhagat, J., Couch, P., Cruickshank, D., Delderfield, M., Dunlop, I., Gamble, M., Michaelides, D., Owen, S., Newman, D., Sufi, S., and Goble, C. (2013).  Why linked data is not enough for scientists.  Future Generation Computer Systems, 29(2):599–611.  Special section: Recent advances in e-Science.
  * Belhajjame et al., (2015) Belhajjame, K., Zhao, J., Garijo, D., Gamble, M., Hettne, K., Palma, R., Mina, E., Corcho, O., Gómez-Pérez, J. M., Bechhofer, S., Klyne, G., and Goble, C. (2015).  Using a suite of ontologies for preserving workflow-centric research objects.  Journal of Web Semantics, 32:16–42.
  * Benda and Engels, (2011) Benda, W. G. and Engels, T. C. E. (2011).  The predictive validity of peer review: A selective review of the judgmental forecasting qualities of peers, and implications for innovation in science.
  * Brack et al., (2020) Brack, A., D’Souza, J., Hoppe, A., Auer, S., and Ewerth, R. (2020).  Domain-independent extraction of scientific concepts from research articles.  In Advances in Information Retrieval, pages 251–266, Cham. Springer International Publishing.
  * Bucur et al., (2019) Bucur, C.-I., Kuhn, T., and Ceolin, D. (2019).  Peer reviewing revisited: Assessing research with interlinked semantic comments.  In In K-CAP 2019: Proceedings of the 10th International Conference on Knowledge Capture, pages 179–187.
  * Bucur et al., (2020) Bucur, C.-I., Kuhn, T., Ceolin, D., and van Ossenbruggen, J. (2020).  A unified nanopublication model for effective and user-friendly access to the elements of scientific publishing.  EKAW2020.
  * Bucur et al., (2021) Bucur, C.-I., Kuhn, T., Ceolin, D., and van Ossenbruggen, J. (2021).  Expressing high-level scientific claims with formal semantics.  In Proceedings of the 11th on Knowledge Capture Conference, K-CAP ’21, page 233–240, New York, NY, USA. Association for Computing Machinery.
  * Chi et al., (2018) Chi, Y., Qin, Y., Song, R., and Xu, H. (2018).  Knowledge graph in smart education: A case study of entrepreneurship scientific publication management.  Sustainability, 10:995.
  * Chibucos et al., (2014) Chibucos, M. C., Mungall, C. J., Balakrishnan, R., Christie, K. R., Huntley, R. P., White, O., Blake, J. A., Lewis, S. E., and Giglio, M. (2014).  Standardized description of scientific evidence using the evidence ontology (eco).  Database : the journal of biological databases and curation.
  * Coulet et al., (2011) Coulet, A., Garten, Y., Dumontier, M., Altman, R. B., Musen, M. A., and Shah, N. H. (2011).  Integration and publication of heterogeneous text-mined relationships on the semantic web.  J Biomed Semant, 2.
  * Etzioni et al., (2005) Etzioni, O., Cafarella, M., Downey, D., Popescu, A.-M., Shaked, T., Soderland, S., Weld, D. S., and Yates, A. (2005).  Unsupervised named-entity extraction from the web: An experimental study.  Artificial Intelligence, 165(1):91–134.
  * Färber and Lamprecht, (2021) Färber, M. and Lamprecht, D. (2021).  The data set knowledge graph: Creating a linked open data source for data sets.  Quantitative Science Studies.
  * Felix and Barrand, (2002) Felix, R. A. and Barrand, M. A. (2002).  P-glycoprotein expression in rat brain endothelial cells: evidence for regulation by transient oxidative stress.  Journal of Neurochemistry, 80:64–72.
  * Garcia-Castro et al., (2013) Garcia-Castro, L., Berlanga, R., Rebholz-Schuhmann, D., and Garcia, A. (2013).  Connections across scientific publications based on semantic annotations.  SEPublica, 10th Extended Semantic Web Conference.
  * Garijo and Poveda-Villalón, (2020) Garijo, D. and Poveda-Villalón, M. (2020).  Best practices for implementing fair vocabularies and ontologies on the web.  In Applications and Practices in Ontology Design, Extraction, and Reasoning.
  * Groth et al., (2010) Groth, P., Gibson, A., and Velterop, J. (2010).  The anatomy of a nanopublication.  Inf. Serv. Use, 30:51–56.
  * Halpin et al., (2010) Halpin, H., Hayes, P. J., McCusker, J. P., McGuinness, D. L., and Thompson, H. S. (2010).  When owl:sameas isn’t the same: An analysis of identity in linked data.  In The Semantic Web – ISWC 2010, pages 305–320, Berlin, Heidelberg. Springer Berlin Heidelberg.
  * Hannestad et al., (2021) Hannestad, L. M., Dancík, V., Godden, M., Suen, I. W., Huellas-Bruskiewicz, K. C., Good, B. M., Mungall, C. J., and Bruskiewich, R. M. (2021).  Knowledge beacons: Web services for data harvesting of distributed biomedical knowledge.  PLoS ONE, 16.
  * Hitzler and van Harmelen, (2010) Hitzler, P. and van Harmelen, F. (2010).  A reasonable semantic web.  Semantic Web, 1:39–44.
  * Hyvönen, (2012) Hyvönen, E. (2012).  Publishing and using cultural heritage linked data on the semantic web.  In Synthesis Lectures on the Semantic Web.
  * Hyvönen, (2020) Hyvönen, E. (2020).  Using the semantic web in digital humanities: Shift from data publishing to data-analysis and serendipitous knowledge discovery.  Semantic Web, 11:187–193.
  * Jacob et al., (2017) Jacob, B., Griffith, D., and Le, T. Q. (2017).  Data.world: A platform for global-scale semantic publishing.  In SEMWEB.
  * Jain et al., (2010) Jain, P., Hitzler, P., Yeh, P. Z., Verma, K., and Sheth, A. P. (2010).  Linked data is merely more data.  In Linked Data Meets Artificial Intelligence, page 82–86. AAAI.
  * Jaradeh et al., (2019) Jaradeh, M. Y., Oelen, A., Farfar, K. E., Prinz, M., D’Souza, J., Kismihók, G., Stocker, M., and Auer, S. (2019).  Open research knowledge graph: Next generation infrastructure for semantic scholarly knowledge.  In Proceedings of the 10th International Conference on Knowledge Capture, K-CAP’19, page 243–246, New York, NY, USA. Association for Computing Machinery.
  * Kotturi et al., (2017) Kotturi, Y., Du, A., Klemmer, S., and Kulkarni, C. (2017).  Long-term peer reviewing effort is anti-reciprocal.  In Proceedings of the Fourth (2017) ACM Conference on Learning @ Scale, L@S ’17, page 279–282, New York, NY, USA. Association for Computing Machinery.
  * Kuhn et al., (2013) Kuhn, T., Barbano, P. E., Nagy, M. L., and Krauthammer, M. (2013).  Broadening the scope of nanopublications.  In Extended Semantic Web Conference, pages 487–5017.
  * Kuhn and Dumontier, (2015) Kuhn, T. and Dumontier, M. (2015).  Making digital artifacts on the web verifiable and reliable.  IEEE Transactions on Knowledge and Data Engineering, 27(9):2390–2400.
  * Kuhn and Dumontier, (2017) Kuhn, T. and Dumontier, M. (2017).  Genuine semantic publishing.  Data Sci., 1:139–154.
  * Kuhn et al., (2021) Kuhn, T., Taelman, R., Emonet, V., Antonatos, H., Soiland-Reyes, S., and Dumontier, M. (2021).  Semantic micro-contributions with decentralized nanopublication services.  PeerJ Computer Science, 7:e387.
  * Lee et al., (2012) Lee, C. J., Sugimoto, C. R., Zhang, G., and Cronin, B. (2012).  Bias in peer review.  Journal of the American Society for Information Science and Technology.
  * Linkov et al., (2006) Linkov, F., Lovalekar, M., and LaPorte, R. (2006).  Scientific journals are ‘’faith based”: is there science behind peer review?  Journal of the Royal Society of Medicine, 99:596–598.
  * Lisena et al., (2019) Lisena, P., Meroño-Peñuela, A., Kuhn, T., and Troncy, R. (2019).  Easy web api development with sparql transformer.  In International semantic web conference, pages 454–470. Springer.
  * Madan et al., (2019) Madan, S., Szostak, J., Elayavilli, R. K., Tsai, R. T.-H., Ali, M., Qian, L., Rastegar-Mojarad, M., Hoeng, J., and Fluck, J. (2019).  The extraction of complex relationships and their conversion to biological expression language (bel) overview of the biocreative vi (2017) bel track.  Database : the journal of biological databases and curation.
  * McGregor, (2008) McGregor, B. (2008).  Facets and hierarchies in scientific search.  The Journal of Electronic Publishing, 11.
  * McNutt et al., (2018) McNutt, M. K., Bradford, M., Drazen, J. M., Hanson, B., Howard, B., Jamieson, K. H., Kiermer, V., Marcus, E., Pope, B. K., Schekman, R., Swaminathan, S., Stang, P., and Verma, I. M. (2018).  Transparency in authors’ contributions and responsibilities to promote integrity in scientific publication.  Proceedings of the National Academy of Sciences of the United States of America, 115:2557 – 2560.
  * Meroño-Peñuela and Hoekstra, (2016) Meroño-Peñuela, A. and Hoekstra, R. (2016).  grlc makes github taste like linked data apis.  In European Semantic Web Conference, pages 342–353. Springer.
  * Papakonstantinou et al., (2018) Papakonstantinou, V., Fundulaki, I., and Flouris, G. (2018).  Assessing linked data versioning systems: The semantic publishing versioning benchmark.  In SSWS@ISWC.
  * Perez-Arriaga, (2018) Perez-Arriaga, M. (2018).  Automated development of semantic data models using scientific publications.
  * Peroni, (2014) Peroni, S. (2014).  The semantic publishing and referencing ontologies.
  * Peroni, (2017) Peroni, S. (2017).  Automating semantic publishing.  Data Sci., 1:155–173.
  * Peroni et al., (2016) Peroni, S., Osborne, F., Iorio, A. D., Nuzzolese, A. G., Poggi, F., Vitali, F., and Motta, E. (2016).  Research articles in simplified html: a web-first format for html-based scholarly articles.  PeerJ Prepr., 4:e2513.
  * Peroni et al., (2013) Peroni, S., Tomasi, F., Vitali, F., and Zingoni, J. (2013).  Semantic lenses as exploration method for scholarly articles.  In T., C., N., F., and A., P., editors, Italian Research Conference on Digital Libraries: Bridging Between Cultural Heritage Institutions, Communications in Computer and Information Science, volume 385 of IRCDL’13, Berlin, Heidelberg, Germany. Springer.
  * Rahardja et al., (2019) Rahardja, U., Lutfiani, N., and Juniar, H. L. (2019).  Scientific publication management transformation in disruption era.  Aptisi Transactions on Management (ATM).
  * Sateli and Witte, (2016) Sateli, B. and Witte, R. (2016).  From papers to triples: An open source workflow for semantic publishing experiments.
  * Senderov and Penev, (2016) Senderov, V. and Penev, L. (2016).  The open biodiversity knowledge management system in scholarly publishing.  Research Ideas and Outcomes, 2.
  * Sernadela et al., (2015) Sernadela, P., Lopes, P., Campos, D., Matos, S., and Oliveira, J. L. (2015).  A semantic layer for unifying and exploring biomedical document curation results.  WBBIO’2015.
  * Shotton, (2009) Shotton, D. M. (2009).  Semantic publishing: the coming revolution in scientific journal publishing.  Learned Publishing, 22.
  * Shotton et al., (2009) Shotton, D. M., Portwin, K., Klyne, G., and Miles, A. J. (2009).  Adventures in semantic publishing: Exemplar semantic enhancements of a research article.  PLoS Computational Biology, 5.
  * Slater, (2014) Slater, T. (2014).  Recent advances in modeling languages for pathway maps and computable biological networks.  Drug Discovery Today, 19.
  * Slater and Song, (2012) Slater, T. and Song, D. H. (2012).  Saved by the bel: ringing in a common language for the life sciences.  Fall.
  * Smith, (1988) Smith, R. (1988).  Problems with peer review and alternatives.  British Medical Journal (Clinical Research Edition), 296(6624):774–777.
  * Smith, (2010) Smith, R. (2010).  Classical peer review: an empty gun.  Breast cancer research, 12.
  * Tenorio-Fornés et al., (2019) Tenorio-Fornés, A., Jacynycz, V., Llop-Vila, D., Sánchez-Ruiz-Granados, A. A., and Hassan, S. (2019).  Towards a decentralized process for scientific publication and peer review using blockchain and ipfs.  In HICSS.
  * Tiddi et al., (2020) Tiddi, I., Balliet, D., and ten Teije, A. (2020).  Fostering scientific meta-analyses with knowledge graphs: A case-study.  The Semantic Web, 12123:287 – 303.
  * Wilkinson et al., (2016) Wilkinson, M. D., Dumontier, M., Aalbersberg, I. J., Appleton, G., Axton, M., Baak, A., Blomberg, N., Boiten, J.-W., da Silva Santos, L. O. B., Bourne, P. E., Bouwman, J., Brookes, A. J., Clark, T., Crosas, M., Dillo, I., Dumon, O., Edmunds, S. C., Evelo, C. T. A., Finkers, R., González-Beltrán, A. N., Gray, A. J. G., Groth, P., Goble, C. A., Grethe, J. S., Heringa, J., ‘t Hoen, P. A. C., Hooft, R. W. W., Kuhn, T., Kok, R. G., Kok, J. N., Lusher, S. J., Martone, M. E., Mons, A., Packer, A. L., Persson, B., Rocca-Serra, P., Roos, M., van Schaik, R. C., Sansone, S.-A., Schultes, E. A., Sengstag, T., Slater, T., Strawn, G. O., Swertz, M. A., Thompson, M., van der Lei, J., van Mulligen, E. M., Velterop, J., Waagmeester, A., Wittenburg, P., Wolstencroft, K., Zhao, J., and Mons, B. (2016).  The fair guiding principles for scientific data management and stewardship.  Scientific Data, 3.
  * Wong, (2019) Wong, C.-Y. (2019).  A century of scientific publication: towards a theorization of growth behavior and research-orientation.  Scientometrics, 119:357–377.
  * Xu et al., (2015) Xu, Y., Mou, L., Li, G., Chen, Y., Peng, H., and Jin, Z. (2015).  Classifying relations via long short term memory networks along shortest dependency paths.  In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 1785–1794, Lisbon, Portugal. Association for Computational Linguistics.
  * Yadav and Bethard, (2018) Yadav, V. and Bethard, S. (2018).  A survey on recent advances in named entity recognition from deep learning models.  In COLING.
  * Zeng et al., (2014) Zeng, D., Liu, K., Lai, S., Zhou, G., and Zhao, J. (2014).  Relation classification via convolutional deep neural network.  In COLING.


qualitative embedding statement of this type was recently established by
Veraar [79, Proposition 3.1].

######  Remark 38.

One can prove a reverse inequality to that of Theorem 36 under the assumption
that $(Y,\|\cdot\|_{Y})$ is a UMD Banach space of type $p\in(1,2]$, in which
case one obtains an upper bound on $\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)}$ in
terms of $\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}$. Specifically, we have the
following estimate for $s>0$.

$\forall f\in
W_{s,p}(\mathbb{R}^{n},Y),\qquad\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)}\lesssim\frac{T_{p}(Y)\beta_{p}(Y)^{4}n^{\frac{s}{2}}}{(p-1)(nV_{n})^{1-\frac{1}{p}}}\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}.$
(130)

Since (130) is not needed below, we omit its proof (which is available on
request). By Remark 33, one can take in (130) $p-1\asymp 1/\beta(Y)$, in which
case $T_{p}(Y)\asymp 1$ and, by (49), $\beta_{p}(Y)\lesssim\beta(Y)^{2}$. Note
that, by Kwapien’s theorem [52], unless $Y$ is isomorphic to a Hilbert space,
its type and cotype do not coincide, and therefore by combining Theorem 36
with (130) one does not obtain an equivalence between the norms
$\|\cdot\|_{H_{s,p}(\mathbb{R}^{n},Y)}$ and
$\|\cdot\|_{W_{s,p}(\mathbb{R}^{n},Y)}$ for non-Hilbertian targets $Y$. See
[38] for a related characterization of Hilbert space.

Recalling the notation that was introduced in Section 2, we end this section
by deducing a corollary of Theorem 36 that will be very important in what
follows.

######  Corollary 39.

Fix $p\in[2,\infty)$ and $(s,\sigma)\in(0,1)\times(1,2)$. Let
$(Y,\|\cdot\|_{Y})$ be a UMD Banach space of cotype $p$. Then every measurable
$f:\mathbb{R}^{n}\to Y$ satisfies

$\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{V_{n}u^{ps+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\lesssim\frac{\sqrt{pn}C_{p}(Y)\beta_{p}(Y)^{4}}{s(1-s)}\|f\|_{H_{s,p}(\mathbb{R}^{n},Y)},$
(131)

and

$\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{V_{n}u^{p\sigma+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\lesssim\frac{\sqrt{p}n^{\frac{3}{2}}C_{p}(Y)\beta_{p}(Y)^{6}}{(\sigma-1)(2-\sigma)}\|f\|_{H_{\sigma,p}(\mathbb{R}^{n},Y)}.$
(132)

###### Proof.

By integrating the conclusion of Corollary 14 (with $q=ps$) over
$x\in\mathbb{R}^{n}$ we see that

$\bigg{(}\frac{1}{V_{n}}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{u^{ps+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\lesssim\frac{\sqrt{pn}}{(nV_{n})^{\frac{1}{p}}}\|f\|_{W_{s,p}(\mathbb{R}^{n},Y)}.$
(133)

A substitution of the conclusion of Theorem 36 into (133) yields (131).

Next, by integrating the conclusion of Corollary 15 (with $q=p\sigma>p$) over
$x\in\mathbb{R}^{n}$ we see that

$\bigg{(}\frac{1}{V_{n}}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\|f^{x}(uy)-P^{1}_{u}f^{x}(uy)\|_{Y}^{p}}{u^{p\sigma+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\lesssim\frac{\sqrt{pn}}{(nV_{n})^{\frac{1}{p}}}\sum_{j=1}^{n}\left\|\frac{\partial
f}{\partial x_{j}}\right\|_{W_{\sigma-1,p}(\mathbb{R}^{n},Y)}.$ (134)

By Theorem 36 applied to $\frac{\partial f}{\partial x_{j}}$ for each
$j\in\\{1,\ldots,n\\}$ and with $s$ replaced by $\sigma-1\in(0,1)$,

$\forall\,j\in\\{1,\ldots,n\\},\qquad\left\|\frac{\partial f}{\partial
x_{j}}\right\|_{W_{\sigma-1,p}(\mathbb{R}^{n},Y)}\lesssim\frac{C_{p}(Y)\beta_{p}(Y)^{4}(nV_{n})^{\frac{1}{p}}}{(\sigma-1)(2-\sigma)}\left\|\frac{\partial
f}{\partial x_{j}}\right\|_{H_{\sigma-1,p}(\mathbb{R}^{n},Y)}.$ (135)

Moreover, by the second assertion of Lemma 35 we have

$\max_{j\in\\{1,\ldots,n\\}}\left\|\frac{\partial f}{\partial
x_{j}}\right\|_{H_{\sigma-1,p}(\mathbb{R}^{n},Y)}\lesssim\beta_{p}^{2}(Y)\left\|f\right\|_{H_{\sigma,p}(\mathbb{R}^{n},Y)}.$
(136)

By substituting (136) into (135), and then substituting the resulting estimate
into (134), we obtain the desired estimate (132). ∎

##  5\. Proof of Theorem 19

We shall prove here the following theorem.

######  Theorem 40.

Fix $p\in[2,\infty)$ and suppose that $(Y,\|\cdot\|_{Y})$ is a UMD Banach
space of cotype $p$. Then every smooth $f:\mathbb{R}^{n}\to Y$ satisfies

$\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{p}}{V_{n}u^{p+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\\\
\lesssim\sqrt{pn}C_{p}(Y)\beta_{p}(Y)^{7}\log(\beta_{p}(Y)n)\sum_{j=1}^{n}\left(\int_{\mathbb{R}^{n}}\Big{\|}\frac{\partial
f}{\partial x_{j}}(x)\Big{\|}_{Y}^{p}\mathrm{d}x\right)^{\frac{1}{p}}.$ (137)

Note that Theorem 19 follows from Theorem 40. Indeed, if $(Y,\|\cdot\|_{Y})$
is a UMD Banach space with $\beta=\beta(Y)$ then by Lemma 32 there exists a
universal constant $\kappa\in(0,\infty)$ such that if we set $p=\kappa\beta$
then $C_{p}(Y)\lesssim 1$ (in particular we necessarily have
$\kappa\beta\geqslant 2$). Moreover, by (49) we have
$\beta_{p}(Y)\lesssim\beta^{2}$. To deduce Theorem 19, take
$f:\mathbb{R}^{n}\to Y$ that is Lipschitz and compactly supported. By
convolving $f$ with a smooth bump function whose support has small diameter we
may also assume that $f$ is smooth. It now follows from Theorem 40 that

$\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{\kappa\beta}}{V_{n}u^{\kappa\beta+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{\kappa\beta}}\\\
\lesssim\beta^{\frac{29}{2}}\sqrt{n}\log(\beta
n)\sum_{j=1}^{n}\left(\int_{\mathbb{R}^{n}}\Big{\|}\frac{\partial f}{\partial
x_{j}}(x)\Big{\|}_{Y}^{\kappa\beta}\mathrm{d}x\right)^{\frac{1}{\kappa\beta}}\leqslant\beta^{15}n^{\frac{5}{2}}\big{(}\mathrm{vol}(\operatorname{supp}(f))\big{)}^{\frac{1}{\kappa\beta}}\|f\|_{\mathrm{Lip}},$

where we used the fact that, due to (8), for every $x\in\mathbb{R}^{n}$ and
$j\in\\{1,\ldots,n\\}$ we have

$\Big{\|}\frac{\partial f}{\partial x_{j}}(x)\Big{\|}_{Y}=\lim_{t\to
0}\frac{\|f(x+te_{j})-f(x)\|_{Y}}{|t|}\leqslant\|f\|_{\mathrm{Lip}}\|e_{j}\|_{X}\stackrel{{\scriptstyle\eqref{eq:john
posiiton}}}{{\leqslant}}\|f\|_{\mathrm{Lip}}\sqrt{n}.$

###### Proof of Theorem 40.

For every $s\in(0,\infty)$ let $\nu_{s}$ be the measure on
$(0,\infty)\times\mathbb{R}^{n}\times B^{n}$ whose density is given by

$\forall(u,x,y)\in(0,\infty)\times\mathbb{R}^{n}\times
B^{n},\qquad\varphi_{s}(u,x,y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{V_{n}|u|^{ps+1}}.$
(138)

Define a linear operator $\mathscr{S}:H_{s,p}(\mathbb{R}^{n},Y)\to
L_{p}(\nu_{s},Y)$ by setting for every $f\in H_{s,p}(\mathbb{R}^{n},Y)$,

$\forall(u,x,y)\in(0,\infty)\times\mathbb{R}^{n}\times
B^{n},\qquad\mathscr{S}f(u,x,y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f^{x}(uy)-P^{1}_{u}f^{x}(uy),$
(139)

where we recall the notations (9) and (15) for $f^{x}$ and $P^{1}_{u}$,
respectively. In what follows we let $M_{s}$ denote the norm of $\mathscr{S}$
as an operator from $H_{s,p}(\mathbb{R}^{n},Y)$ to $L_{p}(\nu_{s},Y)$, i.e.,

$\forall\,s\in(0,\infty),\qquad
M_{s}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\|\mathscr{S}\|_{H_{s,p}(\mathbb{R}^{n},Y)\to
L_{p}(\nu_{s},Y)}.$

Suppose that $s,\sigma,\theta\in\mathbb{R}$ satisfy

$(s,\sigma,\theta)\in(0,1)\times(1,2)\times(0,1)\qquad\mathrm{and}\qquad
1=(1-\theta)s+\theta\sigma.$

Then $\varphi_{1}=\varphi_{s}^{1-\theta}\varphi_{\sigma}^{\theta}$, so by
Stein’s interpolation theorem [71, Theorem 2] for every $f$ in the complex
interpolation space
$[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}$ we have

$\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{p}}{V_{n}u^{p+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\\\
\stackrel{{\scriptstyle\eqref{eq:def density nus}\wedge\eqref{eq:def operator
S}}}{{=}}\|\mathscr{S}f\|_{L_{p}(\nu_{1},Y)}\leqslant
M_{s}^{1-\theta}M_{\sigma}^{\theta}\|f\|_{[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}}.$
(140)

We note that Stein’s interpolation theorem is stated in [71] for real-valued
function spaces, but the standard proofs of this theorem (see also [74] or [9,
Section 5.4]) work without additional effort for vector-valued spaces as well,
which is what we are using here. Alternatively, the vector-valued setting is
treated explicitly by Calderón in [18, Section 13.6]. Every $f\in
W_{1,p}(\mathbb{R}^{n},Y)$ satisfies

$\|f\|_{[H_{s,p}(\mathbb{R}^{n},Y),H_{\sigma,p}(\mathbb{R}^{n},Y)]_{\theta}}\stackrel{{\scriptstyle(*)}}{{\lesssim}}\beta_{p}(Y)\|f\|_{H_{1,p}(\mathbb{R}^{n},Y)}\stackrel{{\scriptstyle(**)}}{{\lesssim}}\beta_{p}(Y)^{3}\|f\|_{W_{1,p}(\mathbb{R}^{n},Y)}.$
(141)

where in $(*)$ we used Lemma 34 and in $(**)$ we used Lemma 35. Hence,

$\bigg{(}\iiint_{\mathbb{R}^{n}\times
B^{n}\times(0,\infty)}\frac{\left\|f^{x}(uy)-P_{u}^{1}f^{x}(uy)\right\|_{Y}^{p}}{V_{n}u^{p+1}}\mathrm{d}x\mathrm{d}y\mathrm{d}u\bigg{)}^{\frac{1}{p}}\\\
\stackrel{{\scriptstyle\eqref{eq:Wp1 def}\wedge\eqref{eq:interpolation ineq
S}\wedge\eqref{eq:use comparisons of
interpolation}}}{{\lesssim}}\beta_{p}(Y)^{3}\bigg{(}\inf_{\stackrel{{\scriptstyle(s,\sigma,\theta)\in(0,1)\times(1,2)\times(0,1)}}{{(1-\theta)s+\theta\sigma=1}}}M_{s}^{1-\theta}M_{\sigma}^{\theta}\bigg{)}\sum_{j=1}^{n}\left(\int_{\mathbb{R}^{n}}\Big{\|}\frac{\partial
f}{\partial x_{j}}(x)\Big{\|}_{Y}^{p}\mathrm{d}x\right)^{\frac{1}{p}}.$ (142)

Corollary 39 asserts that

$\forall(s,\sigma)\in(0,1)\times(1,2),\qquad
M_{s}\lesssim\frac{\sqrt{pn}C_{p}(Y)\beta_{p}(Y)^{4}}{s(1-s)}\qquad\mathrm{and}\qquad
M_{\sigma}\lesssim\frac{\sqrt{p}n^{\frac{3}{2}}C_{p}(Y)\beta_{p}(Y)^{6}}{(\sigma-1)(2-\sigma)}.$

Hence,

$\displaystyle\inf_{\stackrel{{\scriptstyle(s,\sigma,\theta)\in(0,1)\times(1,2)\times(0,1)}}{{(1-\theta)s+\theta\sigma=1}}}M_{s}^{1-\theta}M_{\sigma}^{\theta}$
$\displaystyle\lesssim\inf_{\stackrel{{\scriptstyle(s,\sigma,\theta)\in(0,1)\times(1,2)\times(0,1)}}{{(1-\theta)s+\theta\sigma=1}}}\bigg{(}\frac{\sqrt{pn}C_{p}(Y)\beta_{p}(Y)^{4}}{s(1-s)}\bigg{)}^{1-\theta}\bigg{(}\frac{\sqrt{p}n^{\frac{3}{2}}C_{p}(Y)\beta_{p}(Y)^{6}}{(\sigma-1)(2-\sigma)}\bigg{)}^{\theta}$
$\displaystyle\lesssim\sqrt{pn}C_{p}(Y)\beta_{p}(Y)^{4}\log(\beta_{p}(Y)n),$
(143)

where (5) arises by choosing $s=1-1/\log(\beta_{p}(Y)n)$ and
$\sigma=2-1/\log(\beta_{p}(Y)n)$, in which case necessarily
$\theta=1/\log(\beta_{p}(Y)n)$. We note that these choices essentially yield
the best possible estimate in (5) (up to constant factors), but one could also
choose here, say, $s=1/2$ and $\sigma=3/2$, yielding a worse dependence on $n$
which is of lesser importance for our present purposes. A substitution of (5)
into (142) yields the desired estimate (137). ∎

### Acknowledgements

We are grateful to Charles Fefferman for allowing us to include the contents
of Section 2.2. We also thank Mark Veraar for helpful pointers to the
literature and we are grateful to the anonymous referee for a careful reading
and detailed comments that improved the presentation.

Part of this work was completed during a visit of the three authors to the
MSRI Quantitative Geometry program. Another part of this work was completed
when A. N. was visiting Université Pierre et Marie Curie, Paris, France.

### Added in proof

In the forthcoming work [39], Question 8 is resolved positively by showing
that Theorem 2 holds true for any uniformly convex target $Y$ (in which case
the parameter $\beta$ is replaced by a quantity that depends on the modulus of
uniform convexity of $Y$). Also, the dependence on $n$ that appears in (7) is
improved in [39]. This is achieved in [39] by following the vector-valued
Littlewood–Paley strategy that we introduced here to bound $r^{X\to
Y}(\varepsilon)$, but while implementing it via a method that differs markedly
from the argument that appears in the present work. Specifically, [39] follows
more closely Bourgain’s original strategy [13] for proving his discretization
theorem, though with major differences. In particular, the proof in [39] even
yields a new approach to Dorronsoro’s influential classical work [26] in the
scalar-valued setting.

## References

  * [1] D. J. Aldous.  Unconditional bases and martingales in $L_{p}(F)$.  Math. Proc. Cambridge Philos. Soc., 85(1):117–123, 1979.
  * [2] J. Azzam and R. Schul.  A quantitative metric differentiation theorem.  Proc. Amer. Math. Soc., 142(4):1351–1357, 2014.
  * [3] R. Bañuelos and K. Bogdan.  Lévy processes and Fourier multipliers.  J. Funct. Anal., 250(1):197–213, 2007.
  * [4] R. Bañuelos and P. J. Méndez-Hernández.  Space-time Brownian motion and the Beurling-Ahlfors transform.  Indiana Univ. Math. J., 52(4):981–990, 2003.
  * [5] F. Barthe, O. Guédon, S. Mendelson, and A. Naor.  A probabilistic approach to the geometry of the $l^{n}_{p}$-ball.  Ann. Probab., 33(2):480–513, 2005.
  * [6] S. Bates, W. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman.  Affine approximation of Lipschitz functions and nonlinear quotients.  Geometric and Functional Analysis, 9:1092–1127, 1999.
  * [7] Y. Benyamini.  The uniform classification of Banach spaces.  In Texas functional analysis seminar 1984–1985 (Austin, Tex.), Longhorn Notes, pages 15–38. Univ. Texas Press, Austin, TX, 1985.
  * [8] Y. Benyamini and J. Lindenstrauss.  Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2000.
  * [9] J. Bergh and J. Löfström.  Interpolation spaces. An introduction.  Springer-Verlag, Berlin-New York, 1976.  Grundlehren der Mathematischen Wissenschaften, No. 223.
  * [10] E. Berkson and T. A. Gillespie.  Spectral decompositions and harmonic analysis on UMD spaces.  Studia Math., 112(1):13–49, 1994.
  * [11] J. Bourgain.  Some remarks on Banach spaces in which martingale difference sequences are unconditional.  Ark. Mat., 21(2):163–168, 1983.
  * [12] J. Bourgain.  Vector-valued singular integrals and the $H^{1}$-BMO duality.  In Probability theory and harmonic analysis (Cleveland, Ohio, 1983), volume 98 of Monogr. Textbooks Pure Appl. Math., pages 1–19. Dekker, New York, 1986.
  * [13] J. Bourgain.  Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms.  In Geometrical aspects of functional analysis (1985/86), volume 1267 of Lecture Notes in Math., pages 157–167. Springer, Berlin, 1987.
  * [14] D. Burkholder.  Martingales and singular integrals in Banach spaces.  In Handbook of the geometry of Banach spaces, volume 1, pages 233–269. 2001.
  * [15] D. L. Burkholder.  A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions.  In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pages 270–286. Wadsworth, Belmont, CA, 1983.
  * [16] D. L. Burkholder.  Boundary value problems and sharp inequalities for martingale transforms.  Ann. Probab., 12(3):647–702, 1984.
  * [17] D. L. Burkholder.  Martingales and Fourier analysis in Banach spaces.  In Probability and analysis (Varenna, 1985), volume 1206 of Lecture Notes in Math., pages 61–108. Springer, Berlin, 1986.
  * [18] A.-P. Calderón.  Intermediate spaces and interpolation, the complex method.  Studia Math., 24:113–190, 1964.
  * [19] J. Cheeger.  Quantitative differentiation: a general formulation.  Comm. Pure Appl. Math., 65(12):1641–1670, 2012.
  * [20] J. Cheeger, B. Kleiner, and A. Naor.  Compression bounds for Lipschitz maps from the Heisenberg group to $L_{1}$.  Acta Math., 207(2):291–373, 2011.
  * [21] P. Clément, B. de Pagter, F. A. Sukochev, and H. Witvliet.  Schauder decomposition and multiplier theorems.  Studia Math., 138(2):135–163, 2000.
  * [22] G. David and S. Semmes.  Singular integrals and rectifiable sets in ${\bf R}^{n}$: Beyond Lipschitz graphs.  Astérisque, (193):152, 1991.
  * [23] G. David and S. Semmes.  Analysis of and on uniformly rectifiable sets, volume 38 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1993.
  * [24] G. David and S. Semmes.  Regular mappings between dimensions.  Publ. Mat., 44(2):369–417, 2000.
  * [25] L. Deleaval, O. Guédon, and B. Maurey.  Dimension free bounds for the Hardy–Littlewood maximal operator associated to convex sets.  Preprint, available at http://arxiv.org/abs/1602.02015, 2016.
  * [26] J. Dorronsoro.  A characterization of potential spaces.  Proc. Amer. Math. Soc., 95(1):21–31, 1985.
  * [27] A. Eskin, D. Fisher, and K. Whyte.  Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs.  Ann. of Math. (2), 176(1):221–260, 2012.
  * [28] W. Feller.  An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons Inc., New York, 1971.
  * [29] T. Figiel and G. Pisier.  Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses.  C. R. Acad. Sci. Paris Sér. A, 279:611–614, 1974.
  * [30] T. Figiel and P. Wojtaszczyk.  Special bases in function spaces.  In Handbook of the geometry of Banach spaces, Vol. I, pages 561–597. North-Holland, Amsterdam, 2001.
  * [31] O. Friedland and O. Guédon.  Random embedding of $\ell^{n}_{p}$ into $\ell^{N}_{r}$.  Math. Ann., 350(4):953–972, 2011.
  * [32] D. J. H. Garling.  Brownian motion and UMD-spaces.  In Probability and Banach spaces (Zaragoza, 1985), volume 1221 of Lecture Notes in Math., pages 36–49. Springer, Berlin, 1986.
  * [33] D. J. H. Garling.  Random martingale transform inequalities.  In Probability in Banach spaces 6 (Sandbjerg, 1986), volume 20 of Progr. Probab., pages 101–119. Birkhäuser Boston, Boston, MA, 1990.
  * [34] S. Geiss.  A counterexample concerning the relation between decoupling constants and UMD-constants.  Trans. Amer. Math. Soc., 351(4):1355–1375, 1999.
  * [35] O. Giladi, A. Naor, and G. Schechtman.  Bourgain’s discretization theorem.  Ann. Fac. Sci. Toulouse Math. (6), 21(4):817–837, 2012.
  * [36] S. Guerre-Delabrière.  Some remarks on complex powers of $(-\Delta)$ and UMD spaces.  Illinois J. Math., 35(3):401–407, 1991.
  * [37] R. F. Gundy and N. T. Varopoulos.  Les transformations de Riesz et les intégrales stochastiques.  C. R. Acad. Sci. Paris Sér. A-B, 289(1):A13–A16, 1979.
  * [38] Y.-S. Han and Y. Meyer.  A characterization of Hilbert spaces and the vector-valued Littlewood-Paley theorem.  Methods Appl. Anal., 3(2):228–234, 1996.
  * [39] T. Hytönen and A. Naor.  Heat flow and quantitative differentiation.  Preprint available at https://arxiv.org/abs/1608.01915, 2016.
  * [40] T. Hytönen, J. van Neerven, M. Veraar, and L. Weis.  Analysis in Banach spaces. Vol. I.  Monograph in preparation, draft available at http://fa.its.tudelft.nl/~neerven/ABS/volume1_Hytonen_Neerven_Veraar_Weis.pdf, 2015\.
  * [41] T. P. Hytönen.  Aspects of probabilistic Littlewood-Paley theory in Banach spaces.  In Banach spaces and their applications in analysis, pages 343–355. Walter de Gruyter, Berlin, 2007.
  * [42] T. P. Hytönen.  Littlewood-Paley-Stein theory for semigroups in UMD spaces.  Rev. Mat. Iberoam., 23(3):973–1009, 2007.
  * [43] T. P. Hytönen.  On the norm of the Beurling-Ahlfors operator in several dimensions.  Canad. Math. Bull., 54(1):113–125, 2011.
  * [44] T. Iwaniec and G. Martin.  Riesz transforms and related singular integrals.  J. Reine Angew. Math., 473:25–57, 1996.
  * [45] F. John.  Extremum problems with inequalities as subsidiary conditions.  In Studies and essays presented to R. Courant on his 60th birthday, pages 187–204. Interscience Publishers, Inc., 1948.
  * [46] W. B. Johnson, J. Lindenstrauss, and G. Schechtman.  Banach spaces determined by their uniform structures.  Geom. Funct. Anal., 6(3):430–470, 1996.
  * [47] W. B. Johnson and G. Schechtman.  Embedding $l^{m}_{p}$ into $l^{n}_{1}$.  Acta Math., 149(1-2):71–85, 1982.
  * [48] P. W. Jones.  Square functions, Cauchy integrals, analytic capacity, and harmonic measure.  In Harmonic analysis and partial differential equations (El Escorial, 1987), volume 1384 of Lecture Notes in Math., pages 24–68. Springer, Berlin, 1989.
  * [49] P. W. Jones.  Rectifiable sets and the traveling salesman problem.  Invent. Math., 102(1):1–15, 1990.
  * [50] J.-P. Kahane.  Some random series of functions, volume 5 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, second edition, 1985.
  * [51] I. Karatzas and S. E. Shreve.  Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1991.
  * [52] S. Kwapień.  Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients.  Studia Math., 44:583–595, 1972.  Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI.
  * [53] M. Ledoux and M. Talagrand.  Probability in Banach spaces, volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)].  Springer-Verlag, Berlin, 1991.  Isoperimetry and processes.
  * [54] J. R. Lee and P. Raghavendra.  Coarse differentiation and multi-flows in planar graphs.  Discrete Comput. Geom., 43(2):346–362, 2010.
  * [55] S. Li.  Coarse differentiation and quantitative nonembeddability for Carnot groups.  J. Funct. Anal., 266(7):4616–4704, 2014.
  * [56] S. Li.  Markov convexity and nonembeddability of the Heisenberg group.  To appear in Ann. Inst. Fourier. Available at http://arxiv.org/abs/1404.6751, 2014.
  * [57] S. Li and A. Naor.  Discretization and affine approximation in high dimensions.  Israel J. Math., 197(1):107–129, 2013.
  * [58] M. B. Marcus and G. Pisier.  Characterizations of almost surely continuous $p$-stable random Fourier series and strongly stationary processes.  Acta Math., 152(3-4):245–301, 1984.
  * [59] T. Martínez, J. L. Torrea, and Q. Xu.  Vector-valued Littlewood-Paley-Stein theory for semigroups.  Adv. Math., 203(2):430–475, 2006.
  * [60] J. Matoušek.  On embedding trees into uniformly convex Banach spaces.  Israel J. Math., 114:221–237, 1999.
  * [61] B. Maurey.  Système de Haar.  In Séminaire Maurey-Schwartz 1974–1975: Espaces L$\sup{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. I et II, pages 26 pp. (erratum, p. 1). Centre Math., École Polytech., Paris, 1975.
  * [62] B. Maurey.  Type, cotype and $K$-convexity.  In Handbook of the geometry of Banach spaces, Vol. 2, pages 1299–1332. North-Holland, Amsterdam, 2003.
  * [63] B. Maurey and G. Pisier.  Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach.  Studia Math., 58(1):45–90, 1976.
  * [64] M. Mendel and A. Naor.  Markov convexity and local rigidity of distorted metrics.  J. Eur. Math. Soc. (JEMS), 15(1):287–337, 2013.
  * [65] A. Naor.  Geometric problems in non-linear functional analysis.  Master’s thesis, Hebrew University, 1998.
  * [66] G. Pisier.  Martingales with values in uniformly convex spaces.  Israel J. Math., 20(3-4):326–350, 1975.
  * [67] G. Pisier.  Un exemple concernant la super-réflexivité.  In Séminaire Maurey-Schwartz 1974–1975: Espaces $L^{p}$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, page 12. Centre Math. École Polytech., Paris, 1975\.
  * [68] G. Pisier.  On the dimension of the $l^{n}_{p}$-subspaces of Banach spaces, for $1\leqslant p<2$.  Trans. Amer. Math. Soc., 276(1):201–211, 1983.
  * [69] Y. Qiu.  On the UMD constants for a class of iterated $L_{p}(L_{q})$ spaces.  J. Funct. Anal., 263(8):2409–2429, 2012.
  * [70] G.-C. Rota.  An “Alternierende Verfahren” for general positive operators.  Bull. Amer. Math. Soc., 68:95–102, 1962.
  * [71] E. M. Stein.  Interpolation of linear operators.  Trans. Amer. Math. Soc., 83:482–492, 1956.
  * [72] E. M. Stein.  Singular integrals and differentiability properties of functions.  Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
  * [73] E. M. Stein.  Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton, N.J., 1970.
  * [74] E. M. Stein and G. Weiss.  Interpolation of operators with change of measures.  Trans. Amer. Math. Soc., 87:159–172, 1958.
  * [75] H. Triebel.  Theory of function spaces. II, volume 84 of Monographs in Mathematics.  Birkhäuser Verlag, Basel, 1992.
  * [76] J. M. A. M. van Neerven.  Stochastic evolution equations.  Lecture notes of the Internet Seminar 2007/08 (version May 07, 2010). Available at http://fa.its.tudelft.nl/~neerven/publications/papers/ISEM.pdf, 2010\.
  * [77] J. M. A. M. van Neerven, M. C. Veraar, and L. Weis.  Stochastic integration in UMD Banach spaces.  Ann. Probab., 35(4):1438–1478, 2007.
  * [78] J. M. A. M. van Neerven and L. Weis.  Stochastic integration of functions with values in a Banach space.  Studia Math., 166(2):131–170, 2005.
  * [79] M. Veraar.  Embedding results for $\gamma$-spaces.  In Recent trends in analysis. Proceedings of the conference in honor of Nikolai Nikolski on the occasion of his 70th birthday, Bordeaux, France, August 31 – September 2, 2011, pages 209–219. Bucharest: The Theta Foundation, 2013.