# MeSHup: A Corpus for Full Text Biomedical Document Indexing

###### Abstract

Medical Subject Heading (MeSH) indexing refers to the problem of assigning a
given biomedical document with the most relevant labels from an extremely
large set of MeSH terms. Currently, the vast number of biomedical articles in
the PubMed database are manually annotated by human curators, which is time
consuming and costly; therefore, a computational system that can assist the
indexing is highly valuable. When developing supervised MeSH indexing systems,
the availability of a large-scale annotated text corpus is desirable. A
publicly available, large corpus that permits robust evaluation and comparison
of various systems is important to the research community. We release a large
scale annotated MeSH indexing corpus, MeSHup, which contains 1,342,667 full
text articles in English, together with the associated MeSH labels and
metadata, authors, and publication venues that are collected from the MEDLINE
database. We train an end-to-end model that combines features from documents
and their associated labels on our corpus and report the new baseline.

Keywords: MeSH Indexing, Multi-label text classification

languageresourceLanguage Resources

MeSHup: A Corpus for Full Text Biomedical Document Indexing

Xindi Wang1,3, Robert E. Mercer1,3, Frank Rudzicz2,3,4
---
1Department of Computer Science, University of Western Ontario, London,
Ontario, Canada
2Department of Computer Science, University of Toronto, Toronto, Ontario,
Canada
3Vector Institute, Toronto, Ontario, Canada
4 Unity Health Toronto, Toronto, Ontario, Canada
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

MEDLINE111https://www.nlm.nih.gov/medline/medline_overview.html comprises 33
million (as of Nov. 2021) references to journal articles in the life sciences
with a concentration on biomedicine, which is the National Library of
Medicine’s222https://www.nlm.nih.gov (NLM) premier bibliographic database. It
includes textual information (title and abstract) and bibliographic
information for articles from academic journals covering various disciplines
of the life sciences and biomedicine.
PubMed333https://pubmed.ncbi.nlm.nih.gov/about/ is a free search engine that
provides free access to the MEDLINE database. In addition to MEDLINE, PubMed
also provides access to the PubMed
Central444https://en.wikipedia.org/wiki/PubMed_Central (PMC) repository that
archives open-access, full-text scholarly articles in biomedical and life
sciences journals. All records in the MEDLINE database are indexed with
Medical Subject Headings (MeSH)555https://www.nlm.nih.gov/mesh/meshhome.html –
a controlled and hierarchically-organized vocabulary produced and maintained
by the NLM. As of 2021, there are 29,369 main MeSH headings, and each citation
is indexed with 13 MeSH terms on average. MeSH headings can be further
qualified by 83 subheadings (also known as qualifiers). In addition,
Supplementary Concept Records (SCRs) refer to specific chemical substances in
the MEDLINE records.

MeSH indexing, a process that annotates documents with concepts from
established semantic taxonomies and ontologies, is important for biomedical
text classification and information retrieval. MEDLINE citations are indexed
by human annotators who read the full text of the article and assign the most
relevant MeSH labels to the articles. The manual annotation process ensures
the high quality of indexing but, inevitably, the cost can be prohibitive.
There has been a steady and sizeable increase in the number of citations that
are added to the MEDLINE database every year; for instance, in 2020, 952,919
articles were added (approximately 2,600 on a daily
basis)666https://www.nlm.nih.gov/bsd/medline_pubmed_production_stats.htmland
the average cost of annotation per document is approximately $9.40 [Mork et
al., 2013]. Faced with the growing workload, NLM annotators remain tasked with
indexing newly published articles efficiently and promptly. Therefore, there
is a pressing need for automatic supports to indexing biomedical literature.

Many state-of-the-art models have been proposed to deal with MeSH indexing;
however, there is a clear drawback to these automatic indexing systems because
of the data used to train them. Existing corpora for MeSH indexing only
provide the title and abstract, while human annotators review full text
articles. This suggests that important information might be contained in the
full text which is available to the human indexer, but does not appear in the
title and abstract that automatic indexing systems use to make
recommendations. ?) further indicated that some sections in the full text that
include very specific information, such as the “Methods” section, help improve
the performance of automatic models. Thus, a corpus that contains full text
articles and their associated MeSH labels is highly desirable.

In this work, we construct a new labeled full text MeSH indexing
dataset777https://github.com/xdwang0726/MeSHup, MeSHup, that for each entry,
is a mashup of the PMID, title, abstract, journal, year, author list, MeSH
terms, chemical list, and Supplementary Concept Records from MEDLINE and the
full text introduction, methods, results, discussion, figure captions, and
table captions that are available from BioC-PMC [Comeau et al., 2019]. To the
best of our knowledge, MeSHup is the first publicly available (and the
largest) full text dataset annotated for MeSH indexing. We also propose a
multi-channel model that incorporates extracted features from different
sections of the full text and report its baseline results.

## 2\. Related Work

### 2.1. Biomedical Corpora Related to MeSH Terms

There are several corpora in the biomedical domain that contain or make use of
MeSH terms. The OSHUMED test collection Hersh1994OHSUMEDAI is a set of 348,566
clinically-oriented references in the MEDLINE database which are obtained from
270 medical journals in the years 1987 to 1991. For each citation, the
collection contains the title, abstract, MeSH indexing terms, author, source,
and publication type. The OSHUMED corpus is one of the earliest corpora that
is related to the MeSH indexing task. The GENIA corpus Kim2003GENIAC is a
valuable resource in the biomedical literature that was created to support the
development of tools for text mining and information retrieval and their
evaluation. It contains 2,000 abstracts taken from the MEDLINE database with a
variety of entity types in the GENIA Chemical ontology that are derived from
MeSH terms. The CHEMDNER corpus Krallinger2015TheCC contains 10,000 PubMed
abstracts and 84,355 manually annotated chemical entities. CHEMDNER labels
entities based on the Chemicals and Drugs branch of the MeSH hierarchy and the
MeSH substances. The BioCreative V Chemical-Disease Relation Task Corpus
(BC5CDR) [Li et al., 2015] was developed for the BioCreative V challenge. A
team of Medical Subject Headings (MeSH) indexers for disease/chemical entity
annotation and Comparative Toxicogenomics Database (CTD) curators for CID
relation annotation were invited to ensure high annotation quality and
productivity. Detailed annotation guidelines and automatic annotation tools
were provided. The resulting corpus consists of 1500 PubMed articles with 4409
annotated chemicals, 5818 diseases and 3116 chemical-disease interactions.
Each entity annotation includes both the mention text spans and normalized
concept identifiers, using MeSH as the controlled vocabulary. The NLM-CHEM
corpus [Islamaj et al., 2021], created for Track 2 of BioCreative VII,
consists of 150 full text articles with chemical entity annotations provided
by human experts for 5000 unique chemical names, mapped to 2000 MeSH
identifiers. This dataset is compatible with the CHEMDNER and BC5CDR corpora
described above.

### 2.2. Related Full-Text Biomedical Corpora

A number of full-text corpora have been created in the biomedical domain. Each
has been generated for specific purposes and consequently has been annotated
with that in mind. The earliest is a small five biomedical paper corpus
[Gasperin et al., 2007] which was designed to capture anaphora. The corpus of
biomedical articles are annotated with anaphoric links between coreferent and
associative (biotype, homolog, and set-member) noun phrases referring to the
biomedical entities of interest to the authors. For the BioCreative I task 2
[Hirschman et al., 2005], the training set corpus comprised 803 full text
articles from four different journals that had been previously annotated for
human protein function and the test set corpus comprised 212 full text
articles. The CRAFT (The Colorado Richly Annotated Full-Text) corpus [Verspoor
et al., 2012] is a collection of 97 articles from the PubMed Central Open
Access subset. Each article has been manually annotated along structural,
coreference, and concept dimensions. More recently, the BioC-BioGRID corpus
[Dogan et al., 2017] comprises full text articles annotated for protein-
protein and genetic interactions.

### 2.3. Automatic MeSH Indexing Based on Title and Abstract

As discussed, there is rapid growth in the number of articles in MEDLINE, and
the NLM has developed an indexing tool, Medical Text Indexer (MTI), to
recommend MeSH terms in automated and semi-automated modes [Aronson et al.,
2004]. MTI first takes the title and abstract of the input article and
generates recommended MeSH terms, using a ranking algorithm to determine final
suggestions. There are two important components in MTI: MetaMap Indexing (MMI)
and PubMed-Related Citations (PRC) [Lin and Wilbur, 2007, Aronson and Lang,
2010]. MetaMap recommends MeSH terms based on the mapping of biomedical
concepts in the title and abstract of the input article to the the Unified
Medical Language System888https://www.nlm.nih.gov/research/umls/ (UMLS). PRC
suggests MeSH terms by looking at similar annotations in MEDLINE using
$k$-nearest neighbours. Two sets of recommended MeSH terms are combined to
generate the final MeSH list.

Since 2013, BioASQ999http://bioasq.org[Tsatsaronis et al., 2015] has organized
the biomedical semantic indexing challenge, which offers the opportunity for
more participants to get involved in the MeSH indexing task. BioASQ provides
annotated PubMed articles with the title and abstract only, and participants
can tune their annotation models accordingly. Many effective indexing systems
have been proposed since then, such as MeSHLabeler [Liu et al., 2015],
DeepMeSH [Peng et al., 2016], AttentionMeSH [Jin et al., 2018], MeSHProbeNet
[Xun et al., 2019], and KenMeSH [Wang et al., 2022]. MeSHLabeler and DeepMeSH
are models based on a Learning-to-Rank (LTR) framework. AttentionMeSH and
MeSHProbeNet both utilize deep recursive neural networks (RNNs) and attention
mechanisms, where the main difference is that the former uses label-wise
attention while the latter employs multi-view self-attentive MeSH probes.
KenMeSH combines text features and the MeSH label hierarchy by using a dynamic
knowledge-enhanced mask to index MeSH terms.

### 2.4. MeSH Indexing Based on Full Text

MeSH indexing with full texts has been studied using relatively small sets of
data or restricted to small numbers of specific MeSH terms because of the
limitation of full text access. ?) collected 500 articles in 17 journal issues
in the PubMed database and used the full text as input to MTI. They found that
using the full text of an article provides significantly better (7.4%) quality
of automatic indexing than using only abstracts and titles. ?) used a
collection of 1,413 biomedical articles randomly selected from the PMC Open
Access Subset. They first ran automatic summaries over the full test and then
used the generated summary as input to MTI. The experimental results showed
that incorporating full texts achieved higher (6%) recall with a trade-off in
precision compared to using the abstracts and titles only. ?) collected 14,829
citations and used a rule-based method to classify Check Tags, a small set of
MeSH terms (29 MeSH Check Tags) that represent characteristics of the
subjects. ?) released a full text dataset curated from PMC with 257,590
articles and employed a multi-channel CNN-based feature extraction model.
FullMeSH [Dai et al., 2019] and BERTMeSH [You et al., 2020] used 1.4M full
text articles, the former with an attention-based CNN and the latter with pre-
trained contextual embeddings together with an attention mechanism.
Unfortunately, they did not make their dataset available, which makes it
difficult for others to compare and evaluate their work.

## 3\. Dataset Construction

In this subsection, we introduce how to construct the dataset based on the
PubMed Central Open Access in BioC
format101010https://www.ncbi.nlm.nih.gov/research/bionlp/APIs/BioC-PMC/ (BioC-
PMC) 10.1093/bioinformatics/btz070 and the MEDLINE/PubMed Annual Baseline
Repository111111https://lhncbc.nlm.nih.gov/ii/information/MBR.html (MBR). We
download the entire BioC-PMC subset (as of Nov. 2021) and obtain 3,601,092
full text articles. We also download the entire MBR collection (as of Nov.
2021) and obtain 31,850,051 citations with metadata in the MEDLINE database.
In order to reduce bias, we only consider articles indexed by human annotators
(i.e., articles in the MEDLINE database with modes marked as ‘curated’ (MeSH
terms were provided algorithmically and were human reviewed) or ‘auto’ (MeSH
terms were provided algorithmically) are not
considered)121212https://www.nlm.nih.gov/bsd/licensee/elements_descriptions.html,
and we only focus on articles written in English (i.e., only articles
annotated as ‘eng’ are considered) in the MEDLINE database. We then match
BioC-PMC articles with MBR citations using the PubMed ID (PMID) and obtain a
set of 1,342,667 biomedical documents.

##### Information extracted from BioC-PMC.

Each article in the BioC-PMC subset is structured in a single XML file. The
original published articles are formatted in various ways depending on the
publisher. With the BioC format [Comeau et al., 2019], an article’s textual
information is preserved and each article is organized in a unified structure.
We parse the tags in the BioC formatted XML files to get the section names and
their corresponding texts. We then divide and normalize all full text articles
into eight BioC sections: title, abstract, introduction, methods, results,
discussion, figure captions, and table captions. Table 1 summarizes the
statistical information for the described sections.

Sections | number of articles | average length
---|---|---
Title | 1,342,667 | 16
Abstract | 1,342,667 | 258
Introduction | 1,279,276 | 991
Methods | 1,135,757 | 1446
Results | 1,090,981 | 1640
Discuss | 1,042,379 | 1249
Figure Captions | 1,155,208 | 560
Table Captions | 520,780 | 123

Table 1: Statistics of the generated dataset for each of the eight sections

##### Information extracted from the MEDLINE/PubMed Annual Baseline (MBR).

Starting in 2002, MBR has provided access to all MEDLINE citations and it
updates and adds new citations every year. Each year’s baseline contains
textual information (title and abstract) of the citations as well as various
types of metadata, such as their authors, publishing venues, and references.
Metadata can be regarded as strong indicators for the semantic indexing task
as they include latent information of research topics. We therefore extract
the metadata of each article from MBR; the detailed metadata and their
descriptions are stated in Table 2.

Metadata | Descriptions
---|---
PMID |  The PubMed (NLM database that incorporates MEDLINE) unique identifier.
Authors |  Personal and collective (corporate) author names published with the article.
Journal | The journal that the article is published in.
Year | The year in which the article is published.
DOI | Digital Object Identifiers.
MeSH Terms |  NLM controlled vocabulary, Medical Subject Headings (MeSH).
Supply MeSH | Supplementary Concept Record (SCR) terms.
Chemical List | A list of chemical substances and enzymes.

Table 2: Meta-data extracted from MBR and their descriptions

We combine the information extracted from BioC-PMC and MBR to form MeSHup, a
new large-scale, full-text biomedical semantic indexing dataset. Specifically,
each article in the dataset contains the full textual information and the
metadata associated with it. The keys (section descriptors) from MeSHup are
shown in JSON format in Fig. 1, and an abbreviated version of the dataset is
provided in the Appendix.

⬇

{"articles":[

{"PMID": ,

"TITLE": ,

"ABSTRACT": ,

"INTRO": ,

"METHODS": ,

"RESULTS": ,

"DISCUSS": ,

"FIG_CAPTIONS": ,

"TABLE_CAPTIONS": ,

"JOURNAL": ,

"YEAR": ,

"DOI": ,

"AUTHORS": ,

"MeSH": ,

"CHEMICALS": ,

"SUPPLMeSH":

},

{

...

},

...

]}

Figure 1: The section descriptors given as JSON keys.

Our goal in releasing MeSHup is to promote large-scale ontological
classification of biomedical documents, using full texts, across the
community. With MeSHup, researchers can explore and test state-of-the-art
indexing systems with a common standard.

## 4\. Experiments

Given the full text of a biomedical article, MeSH indexing can be regarded as
a multi-label text classification problem. The learning framework is defined
as follows. $\mathcal{X}=\\{x_{1},x_{2},...,x_{n}\\}$ is a set of biomedical
documents and $\mathcal{Y}=\\{y_{1},y_{2},...,y_{L}\\}$ is the set of MeSH
terms. Multi-label classification studies the learning function
$f:\mathcal{X}\rightarrow[0,1]^{\mathcal{Y}}$ using the training set
$\mathcal{D}=\\{(x_{i},Y_{i}),i=1,...,n\\}$, where $n$ is the number of
documents in the set, and $Y_{i}\subset\mathcal{Y}$ is the set of MeSH labels
for document $x_{i}$. The objective of MeSH indexing is to predict the correct
MeSH labels for any unseen article $x_{k}$, where $x_{k}$ is not in
$\mathcal{X}$.

### 4.1. Baseline Model

In general, traditional MeSH indexing systems focus on the document features
only, thereby suffering from a lack of information about the MeSH hierarchy.
To handle this, we present a hybrid document-label feature method, which is
composed of a multi-channel document representation module, a label feature
representation module, and a classifier. The overall model architecture is
shown is Figure 2.

Figure 2: Model Architecture - There are three main components in our method.
First, a multi-channel document representation module operates on each section
of an input article. Second, a 2-layer GCN creates label vectors. Lastly, a
label-wise attention component calculates the label-specific attention vectors
that are used for predictions.

#### 4.1.1. Multi-channel Document Representation Module

The multi-channel document representation module has five input channels: the
title and abstract channel, the introduction channel, the methods channel, the
results channel, and the discussion channel. Texts in each channel are
represented by the embedding matrices, namely
$E_{\textit{C}}\in\mathbb{R}^{d}$, where $d$ represents the dimension of the
word embeddings.

Instead of a recurrent model which poses computational and technical issues,
we apply a multi-level dilated convolutional neural network (DCNN) to each
channel in order to get the distant effect memory from a CNN model. To be
specific, our DCNN is a three-layer, one-dimensional convolutional neural net
(CNN) with dilated convolution kernels, which obtain high-level semantic
representations of texts without increasing the computation. The concept of
dilated convolution has been popular in semantic segmentation in computer
vision in recent years [Yu and Koltun, 2016, Li et al., 2018], and it has been
introduced to sequential data [Bai et al., 2018], specially to the field of
NLP in neural machine translation [Kalchbrenner et al., 2017] and text
classification [Lin et al., 2018]. Dilated convolution enables exponentially
large receptive fields over the embedding metric, which captures long-term
dependencies over the input texts.

We apply a multi-level DCNN with different dilation rates on top of the
embedding metric on each channel. Larger dilations represent a wider range of
inputs that can capture sentence-level information, whereas small dilations
capture word-level information. The semantic features returned by DCNN for
each channel is denoted as $D_{\textit{C}}\in\mathbb{R}^{(l-s+1)\times 2d}$,
where $l$ is the sequence length in channel $C$ and $s$ is the width of the
convolution kernels.

#### 4.1.2. Label Feature Representation Module

Graph convolutional neural networks (GCNs) [Kipf and Welling, 2017] have
attracted wide attention recently. They have been effective in tasks that have
rich relational structures, as GCNs preserve global information within the
graph. Traditional multi-label text classification mainly focuses on local
consecutive word sequences; for instance, two deep networks commonly used in
building text representations after learning word embeddings are convolutional
(CNNs) and recurrent neural networks (RNNs) [Kim, 2014, Tai et al., 2015].
Some recent studies explored GCN for text classification, where they either
viewed a document or a sentence as a graph of word nodes [Peng et al., 2018,
Yao et al., 2019].

Figure 3: An example of the MeSH hierarchy (retrieved Nov. 2021). For example,
under Body Regions, there are specific body regions under the MeSH tree.

MeSH labels are arrayed hierarchically, an example of which is shown in Figure
3. We take advantage of the structured knowledge we have over the parent and
child relationships in the MeSH label space by using a GCN. In the label graph
setting, we formulate each MeSH label in $\mathcal{Y}$ as a node in the graph.
The edges represent parent and child relationships among the MeSH terms. The
edge types of a node contain edges from itself, from its parent, and from its
children. We first take the average of the word embeddings of the words in the
MeSH label descriptor as the initial label embedding
$v_{\textit{i}}\in\mathbb{R}^{d}$:

$v_{i}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}w_{j},i=1,2,...,L,$ (1)

where $m_{i}$ is the number of words in the descriptor of label $i$,
$w_{\textit{j}}$ is the word embedding of word $j$, and $L$ is the number of
labels. The GCN layers capture information about immediate neighbours with one
layer of convolution, and information from a larger neighbourhood can be
integrated by using a multi-layer stack. We use a two-layer GCN to incorporate
the hierarchical information among MeSH terms. At each GCN layer, we aggregate
the parent and child nodes for the $i^{th}$ label to form the new label
embedding for the next layer:

$h^{l+1}=\sigma(A\cdot h^{l}\cdot W^{l}),$ (2)

where $h^{l}$ and $h^{l+1}\in\mathbb{R}^{L\times d}$ indicate the node
presentations of the $l^{th}$ and $(l+1)^{th}$ layers, $\sigma(\cdot)$ denotes
an activation function, $A$ is the adjacency matrix of the MeSH hierarchical
graph, and $W^{l}$ is a layer-specific trainable weight matrix. Next, we sum
both the averaged description vector from Equation 1 with the GCN output to
form:

$H_{\textit{label}}=v+h^{l+1},$ (3)

where $H_{\textit{label}}\in\mathbb{R}^{L\times d}$ is the final label matrix.

#### 4.1.3. Classifier

For large multi-label text classification tasks, relevant information for each
label might be scattered in various locations of the article. In order to
match documents to their corresponding label vectors, we employ label-wise
attention following ?). We generate label-specific representations for each
channel:

$\begin{gathered}\alpha_{\textit{C}}=\textrm{Softmax}(D_{\textit{C}}\cdot
H_{\textit{label}})\\\
\textit{content}_{\textit{C}}=\alpha_{\textit{C}}^{T}\cdot
D_{\textit{C}},\end{gathered}$ (4)

where $\textit{content}_{\textit{C}}\in\mathbb{R}^{L\times d}$. We then sum up
the label-specific content representation for each channel to form the final
document representation for each article:

$D=\sum\textit{content}_{\textit{C}}$ (5)

We then generate the predicted score for each MeSH label via:

$\hat{y_{i}}=\sigma(D\odot H_{\textit{label}}),i=1,2,...,L,$ (6)

where $\sigma(\cdot)$ denotes the sigmoid function. Training our proposed
method uses binary cross-entropy as the loss function:

$L=\sum_{i=1}^{L}[-y_{i}\cdot\log(\hat{y_{i}})-(1-y_{i})\cdot\log(1-\hat{y_{i}}))],$
(7)

where $y_{i}\in[0,1]$ is the ground truth of label $i$, and
$\hat{y_{i}}\in[0,1]$ denotes the prediction of label $i$ obtained from the
proposed model.

### 4.2. Evaluation Metrics

We evaluate performance of MeSH indexing systems using two groups of
measurements: bipartition-based evaluation and ranking-based evaluation.
Bipartition evaluation is further divided into example-based and label-based
metrics. Example-based measures, computed per data point, compute the harmonic
mean of standard precision (EBP) and recall (EBR) for each data point. The
metrics are defined as:

$\textit{EBF}=\frac{2\times\textit{EBR}\times\textit{EBP}}{\textit{EBR}+\textit{EBP}},$
(8)

where

$\textit{EBP}=\frac{1}{N}\sum_{i=1}^{N}\frac{|y_{i}\cap\hat{y_{i}}|}{|\hat{y_{i}}|},$
(9)
$\textit{EBR}=\frac{1}{N}\sum_{i=1}^{N}\frac{|y_{i}\cap\hat{y_{i}}|}{|y_{i}|},$
(10)

where $y_{i}$ is the true label set and $\hat{y_{i}}$ is the predicted label
set for instance $i$, and $N$ represents the total number of instances.

We perform label-based evaluation for each label in the label set. The
measurements include Micro-average F-measure (MiF) and Macro-average F-measure
(MaF). MiF aggregates the global contributions of all MeSH labels and then
calculates the harmonic mean of micro-average precision (MiP) and micro-
average recall (MiR), which are heavily influenced by frequent MeSH terms. MaF
computes the macro-average precision (MaP) and macro-average recall (MaR) for
each label and then averages them, which provides equal weight to each MeSH
term. Therefore frequent MeSH terms and infrequent ones are equally important.
The aforementioned metrics are defined as follows:

$\textit{MiF}=\frac{2\times\textit{MiR}\times\textit{MiP}}{\textit{MiR}+\textit{MiP}},$
(11)

where

$\textit{MiP}=\frac{\sum_{j=1}^{L}\textit{TP}_{j}}{\sum_{j=1}^{L}\textit{TP}_{j}+\sum_{j=1}^{L}\textit{FP}_{j}},$
(12)
$\textit{MiR}=\frac{\sum_{j=1}^{L}\textit{TP}_{j}}{\sum_{j=1}^{L}\textit{TP}_{j}+\sum_{j=1}^{L}\textit{FN}_{j}},$
(13)

$\textit{MaF}=\frac{2\times\textit{MaR}\times\textit{MaP}}{\textit{MaR}+\textit{MaP}},$
(14)

where

$\textit{MaP}=\frac{1}{L}\sum_{j=1}^{L}\frac{\textit{TP}_{j}}{\textit{TP}_{j}+\textit{FP}_{j}},$
(15)
$\textit{MaR}=\frac{1}{L}\sum_{j=1}^{L}\frac{\textit{TP}_{j}}{\textit{TP}_{j}+\textit{FN}_{j}},$
(16)

where $\textit{TP}_{j}$, $\textit{FP}_{j}$ and $\textit{FN}_{j}$ are the true
positives, false positives, and false negatives, respectively, for each label
$l_{j}$ in the set of all labels $L$.

Ranking-based evaluation includes precision at k (P@k) and recall at k (R@k).
P@k shows the number of relevant MeSH terms that are suggested in the top-$k$
recommendations of the MeSH indexing system, and R@k indicates the proportion
of relevant items that are suggested in the top-$k$ recommendations. The
metrics are defined as follows:

$\textit{P@k}=\frac{1}{k}\sum_{l\in r_{k}\left(\hat{y}\right)}y_{l},$ (17)
$\textit{R@k}=\frac{1}{|y_{i}|}\sum_{l\in r_{k}\left(\hat{y}\right)}y_{l},$
(18)

where $r_{k}$ returns the top-$k$ recommended items.

Thresholds greatly impact the bipartition-based evaluation metrics. We
therefore tuned the threshold $\tau_{i}$ for predicting the $i$-th label, and
selected the predicted MeSH term (MeSHi) whose predicted probability is
greater than $\tau_{i}$:

$\textit{MeSH}_{i}=\begin{cases}\hat{y_{i}}\geq\tau_{i},1\\\
\hat{y_{i}}<\tau_{i},0\end{cases}$ (19)

We used the micro-F optimization algorithm proposed by ?) to tune the
thresholds:

$\tau_{i}=\operatorname*{argmax}_{\mathcal{T}}\textit{MiF}(\mathcal{T}),$ (20)

where $\mathcal{T}$ represents all possible threshold values for label $i$.

### 4.3. Experiment Settings

We implement our model using PyTorch [Paszke et al., 2019]. We use
200-dimensional word embeddings (BioWordVec) that are pretrained on PubMed
article titles and abstracts [Zhang et al., 2019]. For the model’s DCNN
component, we use a 1-dimension convolution with kernel size 3 and a three-
level dilated convolution with dilation rates [1,2,3]. The number of hidden
units in both components of our model is set to 200. We use the Adam optimizer
[Kingma and Ba, 2015] with a minibatch size of 8 and an initial learning rate
of 0.0003 with a decay rate of 0.9 in every epoch. To avoid overfitting, we
apply dropout directly after the embedding layer with a rate of 0.2 and use
early stopping strategies [Yao et al., 2007]. Our model is trained on a single
NVIDIA A100 GPU. It takes approximately five to seven days to train the full
model.

### 4.4. Experimental Results

In the baseline model, we are interested in the articles that have all six
sections, i.e., title, abstract, introduction, method, results, and discuss.
We extract the 957,426 articles from MeSHup that meet these criteria. We use
stratified sampling over publication year to split our dataset into training,
validation and testing. We use 80% of the documents for training (765,920),
10% for validation (95,737), and 10% for testing (95,769).

We first conduct our experiments with titles and abstracts only, and then we
do our experiments on the full texts. From this experiment, we would like to
see how integrating full text information affects the indexing performance
compared with using the titles and abstracts only. Table 3 summarizes the
results of bipartition evaluation and Table 4 shows the results of ranking-
based measures. We can see substantial improvements on all evaluation metrics
when involving full texts, which indicates that full texts are more
informative compared to the titles and abstracts. The baseline model preforms
fairly well on precisions, but with a trade off in recalls. The reason for
this could be that the frequency of each MeSH label is quite biased, some
labels might have very few training examples so that the baseline model is
very hard to predict those rare labels.

Bipartition evaluation | Methods
---|---
|  | Titles and Abstracts | Full Texts
Example based | EBF | 0.183 | 0.259
EBP | 0.503 | 0.588
EBR | 0.112 | 0.166
Micro-averaged | MiF | 0.177 | 0.259
MiP | 0.473 | 0.604
MiR | 0.110 | 0.164
Macro-averaged | MaF | 0.362 | 0.367
MaP | 0.798 | 0.810
MaR | 0.234 | 0.237

Table 3: Comparison using only titles and abstracts and full texts across
bipartition evaluation. Bold: best scores in each row.

Ranking Based | Methods
---|---
Measure
|  | Titles and Abstracts | Full Texts
P@k | $P@1$ | 0.699 | 0.801
$P@3$ | 0.462 | 0.609
$P@5$ | 0.372 | 0.496
$P@10$ | 0.260 | 0.341
$P@15$ | 0.205 | 0.267
R@k | $R@1$ | 0.051 | 0.077
$R@3$ | 0.098 | 0.128
$R@5$ | 0.131 | 0.171
$R@10$ | 0.180 | 0.232
$R@15$ | 0.214 | 0.272

Table 4: Comparison using only titles and abstracts and full texts across
ranking-based measures. Bold: best scores in each row.

## 5\. Conclusion and Future Work

We present MeSHup, a new, publicly available full text dataset annotated for
MeSH indexing. It is a mashup of full-text information from BioC-PMC and
associated metadata collected from the MEDLINE database. This is the first
large dataset that contains the full text information that allows the research
community to incorporate more textual information other than the title and
abstract in building MeSH indexing systems. We also train an end-to-end model
that comprises features extracted from the document itself and features
obtained from labels.

We think that the MeSHup dataset could be a valuable resource not only for
MeSH indexing but also for full text mining and retrieval. Since our analysis
covers several but not all sections of the full text articles, it is likely
that other parts of the article together with metadata may also have impacts
on future outcomes. In future, we plan to involve more sections of the textual
information and metadata to improve the automatic indexing system.

## Acknowledgements

We thank all reviewers for their constructive comments and feedback. Resources
used in preparing this research were provided, in part, by Compute
Ontario131313https://www.computeontario.ca, Compute
Canada141414https://www.computecanada.ca, the Province of Ontario, the
Government of Canada through CIFAR, and companies sponsoring the Vector
Institute151515https://www.vectorinstitute.ai/partners. This research is
partially funded by The Natural Sciences and Engineering Research Council of
Canada (NSERC) through a Discovery Grant to R. E. Mercer. F. Rudzicz is
supported by a CIFAR Chair in AI.

## Appendix: A Complete Version of A Data Sample in the Dataset

⬇

{"articles":[

{"PMID":"27976717",

"TITLE":"Temporal pairwise spike correlations fully capture single-neuron
information",

"ABSTRACT":"To crack the neural code and read out the information neural
spikes convey, [...]",

"INTRO":"Throughout the central nervous system of a mammalian brain [...]",

"METHODS":"Deriving the correlation theory of neural information [...]",

"RESULTS":"We are interested in the information contained in a spike train
r(t) about a stimulus s(t)[...]",

"DISCUSS":"The list of spike timing features that have been implicated in
neural coding includes [...]",

"FIG_CAPTIONS":"Dimensionality of neural information coding [...]",

"TABLE_CAPTIONS":"Parameter sets across neuron models. [...]",

"JOURNAL":"Nature communications",

"YEAR":"2016",

"DOI":"10.1038/ncomms13805",

"AUTHORS":[

"Amadeus,Dettner",

"Sabrina,Munzberg",

"Tatjana,Tchumatchenko"],

"MeSH": {

"D000200":"Action Potentials",

"D008959":"Models, Neurological",

"D009474":"Neurons",

"D059010":"Single-Cell Analysis"

},

"CHEMICALS":"None",

"SUPPLMeSH":"None"

},

{

...

},

...

]}

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## 7\. Language Resource References

lrec2022-bib languageresource


# Thermal noise competes with turbulent fluctuations below millimeter scales

Dmytro Bandak  Department of Physics, University of Illinois at Urbana-
Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana,
Illinois 61801, USA  Gregory L. Eyink  Department of Applied Mathematics and
Statistics, and Department of Physics, The Johns Hopkins University,
Baltimore, MD, USA, 21218  Alexei Mailybaev  Instituto de Matemática Pura e
Aplicada - IMPA, Rio de Janeiro, Brazil  Nigel Goldenfeld  Department of
Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of
Physics, 1110 West Green Street, Urbana, Illinois 61801, USA

###### Abstract

Turbulent flows frequently accompany physical, chemical and biological
processes, such as mixing, two-phase flow, combustion and even foraging by
bacteria and plankton larvae, all of which are in principle subject to thermal
fluctuations already on scales of several microns. Nevertheless the large
separation between the millimeter scale at which turbulent fluctuations begin
to be strongly damped and the mean free path of the fluid has been generally
assumed to imply that thermal fluctuations are irrelevant to the turbulent
dissipation range. Here we use statistical mechanical estimates to show that
thermal fluctuations are not negligible compared to turbulent eddies in the
dissipation range. Simulation of the Sabra shell model shows that intermittent
bursts of turbulence lead to a fluctuating length scale below which thermal
fluctuations are important: over three decades of length, from sub-millimeter
scales down to the mean free path, thermal fluctuations coexist with
hydrodynamics. Our results imply that thermal fluctuations cannot be neglected
when modeling turbulent phenomena in the far dissipation range.

Thermal effects are known to play a prominent role in many physical, chemical
and biological processes in molecular fluids. These include high
Schmidt/Prandtl-number scalar mixing Donev  _et al._ (2014a), droplet and
bubble formation Chaudhri  _et al._ (2014); Gallo  _et al._ (2020), locomotion
of micro-organisms Götze and Gompper (2010), combustion Lemarchand and
Nowakowski (2004); Bhattacharjee  _et al._ (2015), and others. Often the
relevant flows are turbulent, in which case all current approaches to model
these processes — high Schmidt-number scalar mixing Clay  _et al._ (2018);
Buaria  _et al._ (2020), droplet and bubble formation Elghobashi (2019); Milan
_et al._ (2020), cellular motility Durham  _et al._ (2013); Wheeler  _et al._
(2019), and combustion Donzis  _et al._ (2010); Sreenivasan (2004); Driscoll
(2008); Echekki and Mastorakos (2010) — focus on turbulent fluctuations due to
fluid inertia damped by viscosity and ignore thermal effects completely. This
neglect is commonly justified by the idea that there is a strong separation
between hydrodynamic scales dominated by turbulent fluctuations and extremely
small scales of order the mean-free path length where thermal fluctuations
begin to play a role Von Neumann (1963). The smallest turbulent fluid scales
below which eddies are damped by viscosity, known as the dissipation range,
typically occurs at millimeter scales and there is currently intensive effort,
by computation Khurshid  _et al._ (2018); Gorbunova  _et al._ (2020); Buaria
and Sreenivasan (2020), theory Pauls and Ray (2020); Canet  _et al._ (2017);
Gibbon and Dubrulle (2021) and experiment Debue  _et al._ (2018); Gorbunova
_et al._ (2020); Debue  _et al._ (2021), to understand the dynamics and
statistics at these scales. The motivation ranges from unraveling the nature
of turbulence itself, and the associated unsolved issue of formation of
singularities Fefferman (2006), to the phenomena and practical applications
mentioned above and others.

The basic underlying assumption in turbulent flows of a viscous dissipation
range well-separated from thermal fluctuations is at first sight surprising.
This seemingly contradicts the fluctuation-dissipation theorem of statistical
physics (e.g. see de Zarate and Sengers (2006)), which implies that thermal
fluctuations and dissipation should be intrinsically tied together, even in a
far from equilibrium system where local thermal equilibrium can be justified.
The neglect of thermal effects in the dissipation range is convenient; it
allows one to use the Navier-Stokes partial differential equations exclusively
to model the phenomena of interest and even this requires a computational tour
de force for turbulent flows Khurshid  _et al._ (2018); Gorbunova  _et al._
(2020); Buaria and Sreenivasan (2020). On the other hand, the hydrodynamic
equations consistent with the fluctuation-dissipation theorem for fluids
locally in thermal equilibrium were formulated already by Landau and Lifshitz
in 1959 Landau and Lifshitz (1959); de Zarate and Sengers (2006), and the
effects of thermal fluctuations measured at the onset of Rayleigh-Bénard
convection Wu  _et al._ (1995). As far as turbulence is concerned, most prior
discussions Hosokawa (1976); Ruelle (1979); Macháček (1988) suggest that
thermal noise plays only only a secondary role in the selection and uniqueness
of the stationary measures characterizing turbulence. To what extent are these
seemingly abstract considerations relevant to real flows, especially given the
large separation in scales between the mean free path and those where
hydrodynamic dissipation occurs?

The purpose of this Letter is to provide theoretical arguments and numerical
evidence that thermal noise effects manifest throughout the turbulent
dissipation range, in contradiction to widespread belief. Our results imply
that a nontrivial interplay between turbulent and thermal fluctuations must
occur at small scales for nearly all flows in Nature and in the laboratory.
Our results support pioneering ideas of Betchov Betchov (1957, 1961, 1964),
who reached similar conclusions already in 1957, and extend them to account
for the effects of inertial-range intermittency.

In the low Mach number ($Ma$) limit, fluctuating hydrodynamics adds a
stochastic stress term
${\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$\tau$}}$ to the
incompressible Navier-Stokes equation for the velocity ${\bf u}$ satisfying
Forster  _et al._ (1976, 1977); Usabiaga  _et al._ (2012); Donev  _et al._
(2014b) ${\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\cdot$}}{\bf u}=0$ and

$\partial_{t}{\bf u}+({\bf u}\cdot{\mbox{\boldmath$\nabla$}}){\bf
u}=-{\mbox{\boldmath$\nabla$}}p+\nu\Delta{\bf
u}+{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$\tau$}}.$ (1)

In accord with the general fluctuation-dissipation relation, the stress
$\tau_{ij}({\bf x},t)$ which models thermal fluctuations is a Gaussian random
field with mean zero and covariance

$\displaystyle\langle\tau_{ij}({\bf x},t)\tau_{kl}({\bf
x}^{\prime},t^{\prime})\rangle$ $\displaystyle=$ $\displaystyle\frac{2\nu
k_{B}T}{\rho}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl}\right)$
(3) $\displaystyle\hskip 30.0pt\times\delta^{3}({\bf x}-{\bf
x}^{\prime})\delta(t-t^{\prime}),$

which is proportional to the kinematic viscosity $\nu,$ the dissipative
transport coefficient of the fluid, and absolute temperature $k_{B}T$ in
energy units set by Boltzmann’s constant $k_{B}$. The other parameter
appearing in (3) is $\rho,$ the mass density of the fluid. It is precisely
Eq.(1) that one must solve numerically Usabiaga  _et al._ (2012); Donev  _et
al._ (2014b), or otherwise, in order to model molecular fluids at low Mach
number.

Noise estimate. We now rehearse standard arguments Von Neumann (1963) that
thermal effects in fluids become important only at scales comparable to the
mean free path length $\lambda_{mfp}$, and then explain why these arguments
are not valid. The velocity fluctuations at length scale $\ell$ from thermal
fluctuations can be estimated as

$u_{\ell}^{th}\sim c_{th}/(n\ell^{3})^{1/2}=c_{th}(\ell_{int}/\ell)^{3/2},$
(4)

where $c_{th}$ is the speed of sound, $n$ is the number density and
$\ell_{int}$ is the typical interparticle distance. The standard argument asks
when thermal fluctuations at the scale $\ell$ are of the same order as a
typical flow velocity $u$; equating them, we find that
$\ell\sim\ell_{int}(Ma)^{-2/3}$, where $Ma=u/c_{th}$ is the corresponding Mach
number. By this argument, thermal fluctuations become important only at scales
comparable to $\ell_{int}$, which is of the same order as $\lambda_{mfp}$ in
liquids and $\ll\lambda_{mfp}$ in gases. Even for $Ma\ll 1,$ the thermal
fluctuation velocities are still much smaller than $u$ until $\ell$ becomes
comparable to $\ell_{int}$, and so can be neglected. The flaw in this argument
is that the thermal velocity at scale $\ell$ should not be compared with the
mean velocity or r.m.s. velocity $u$ of the eddies of largest size $L$.
Instead it is more appropriate to compare $u^{th}_{\ell}$ with the r.m.s
velocity $u_{\ell}$ of turbulent eddies at the same scale $\ell$. In the
dissipation range $u_{\ell}$ drops exponentially with decreasing $\ell$ so
that thermal fluctuations very quickly become competitive with turbulent
fluctuations, as is shown by the following quantitative estimate.

Figure 1: Plot of the model turbulent energy spectrum (5) for the parameters
of the atmospheric boundary layer (7) and for a typical Reynolds number
$Re=10^{7}$, as a solid line. The dashed line is the spectrum with no thermal
noise.

We consider a standard model of the turbulent energy spectrum devised by von
Kármán Von Kárḿan (1948), supplemented with an exponential factor to represent
decay of turbulent fluctuations due to viscous effects. The estimate (4) of
thermal velocity fluctuations can be equivalently restated as a $k^{2}$
equipartition energy spectrum, as first noted for equilibrium fluids in 3D by
Lee Lee (1952), and Hopf Hopf (1952). The entire model spectrum then becomes

$E(k)=C_{K}(\varepsilon
L)^{2/3}\frac{L^{5}k^{4}}{(1+(kL)^{2})^{17/6}}\exp(-\beta k\eta)+Ak^{2},$ (5)

where $C_{K}$ is the Kolmogorov constant, $\beta$ is the exponential decay
rate in the far dissipation range and
$A=\frac{4\pi}{(2\pi)^{3}}\frac{k_{B}T}{\rho}$ is the factor associated to
thermal fluctuations in 3D. This model is plotted in Fig.1 with parameter
values appropriate to the atmospheric boundary layer (ABL) Garratt (1994)

$\displaystyle L=10^{5}\ cm,\quad\varepsilon=400\
cm^{2}/sec^{3},\quad\nu=0.15\ cm^{2}/sec,$ (6) $\displaystyle\rho=1.2\times
10^{-3}\ g/cm^{3},\quad T=300^{\circ}K,$ (7)

and with typical values of $\beta=7$ Khurshid  _et al._ (2018) and $C_{K}=1.6$
Donzis and Sreenivasan (2010). One can see in Fig.1 the familiar regimes of
turbulent flow, the inertial range with $E(k)\propto k^{-5/3}$, which
transitions into the dissipation range with $E(k)\propto\exp(-\beta k\eta)$
around the Kolmogorov scale $\eta=(\nu^{3}/\epsilon)^{1/4}\sim 0.5\
\mathrm{mm}$. However, the model (5) predicts that once thermal fluctuations
are included, this exponential decay is completely replaced by the $k^{2}$
equipartition spectrum. The presence of this thermal component on top of the
turbulent spectrum does not automatically follow from the inclusion of thermal
fluctuations in the dynamical equations (1), but can arise through the
separation of time-scales, if it turns out that viscosity and thermal noise
are more dominant than the nonlinear term in (1) for dissipation-range scales.
Thus it is necessary to study this issue quantitatively, which we do below. We
may note that thermal fluctuations have been considered already in numerical
simulations of superfluid turbulence using Gross-Pitaevskii equation Shukla
_et al._ (2019), and similar equipartition spectra observed.

We can estimate on order of magnitude the wavenumber $k_{eq}$ at which
exponential decay is overtaken by thermal equipartition just by equating the
two spectra, as
$u_{\eta}^{2}\eta\mathrm{e}^{-k\eta}\sim\frac{k_{B}T}{\rho}k^{2}$ where
$u_{\eta}=(\varepsilon\nu)^{1/4}$ is the Kolmogorov velocity. This leads to a
formula for $k_{eq}$ of the form

$(k_{eq}\eta)^{2}\mathrm{e}^{k_{eq}\eta}=1/\theta_{K}$ (8)

in which appears the dimensionless ratio

$\theta_{K}=\frac{k_{B}T}{\rho u_{\eta}^{2}\eta^{3}}.$ (9)

The quantity $\theta_{K}$ is a new dimensionless number group that
characterizes the turbulent flow, in addition to the Reynolds number $Re.$
Substituting into (8) the values of parameters (7) characteristic of the ABL,
we arrive at the value $\theta_{K}=2.83\times 10^{-8}.$ It is then easy to
calculate from (8) that $k_{eq}\eta\doteq 11,$ which in the ABL corresponds to
a length $\ell_{eq}\simeq 49\ \mu$m. In fact, due to the rapid increase of the
exponential in (8), this equipartition scale $\ell_{eq}$ is very insensitive
to the precise value of $\theta_{K}$ and always takes a value only a factor of
10 or so smaller than $\eta.$ This result is strikingly different from the
naive expectation that thermal effects become important only at length scales
comparable to the mean free path, which is $\lambda_{mfp}=68\
\mathrm{nm}\sim\eta/10^{4}$ in the ABL. Alternatively, using parameters from
water experiments of Debue  _et al._ (2018), $\eta=16\ \mu\mathrm{m}$ and
$\theta_{K}\doteq 2.5\times 10^{-7},$ we estimate that thermal fluctuations
should be significant from just below $16\ \mu\mathrm{m}$ down to
$\lambda_{mfp}\doteq 0.25\ \mathrm{nm}$. In summary, naturally occurring and
practically relevant turbulent flows possess several decades of length scales
where a hydrodynamic approximation is justified and yet fluctuations are
dominated by thermal effects.

The previous arguments and conclusions are nearly identical to those presented
by Betchov Betchov (1957), except that, rather than an energy spectrum
exhibiting exponential decay, he assumed a fast power-law decay $\sim k^{-n}$
with $n\doteq 6$-$7$ in the far-dissipation range, as predicted by
contemporary theories of Heisenberg Heisenberg (1948) and others. To assess
these arguments and to extend them beyond the energy spectrum to higher-order
moments sensitive to intermittency effects, we now consider a simple dynamical
model of turbulence which is believed to capture the nontrivial multifractal
scaling properties of turbulence.

Figure 2: The energy spectrum of the deterministic model (\-----), noisy model
(\-----), and the thermal equipartition value $2\theta_{K}$ (\- - -). Inset:
The local stretching exponents (11) for the deterministic model spectrum
($\bullet$) and its theoretical value in (10) (\- - -).

Numerical simulations. To further understand the interplay of turbulent and
thermal fluctuations we turn to numerical simulations. Unfortunately, direct
numerical simulations of the fluctuating Navier-Stokes equation (1) with
sufficient range of scales to study how rare intense events at high Reynolds
number interact with thermal fluctuations are not yet feasible, even with the
largest supercomputers in the world. We therefore resort to shell models,
simplified dynamical systems that are widely used as surrogates of Navier-
Stokes equations in order to perform high Reynolds number simulations Biferale
(2003) of scaling properties. In particular, we employ the Sabra shell model
L’vov  _et al._ (1998) supplemented with stochastic terms controlled by
”temperature“ to represent thermal noise. It is defined by a set of coupled
nonlinear stochastic ODEs that describe evolution of complex variables $u_{n}$
defined at discrete wavenumbers $k_{n}=k_{0}2^{n},$ $n=0,1,2,...,N$. Here
$u_{n}$ represents the “velocity” of an eddy of size $1/k_{n}$. At large times
the Sabra model with fixed forcing attains a statistical steady-state with
constant flux of energy
$E=\sum_{n}\epsilon_{n}=\frac{1}{2}\sum_{n}|u_{n}|^{2}$ due to the cascade
across scales in the inertial range. To understand the relative importance of
turbulent and thermal fluctuations, we simulated the system for times large
enough to represent the steady-state with the parameters in (7) for the ABL,
achieving the Reynolds number $Re\sim 10^{6}$. The simulation was done twice,
once with thermal noise, once without. Further details on stochastic Sabra
model and our numerical stochastic integration scheme, as well as convergence
tests can be found in Supplementary information and elsewhere Eyink  _et al._
(2021).

Discussion of results. The numerical results confirm our basic claim that
thermal equipartition is achieved at scales close to $\eta/10$. This can be
readily seen in Fig. 2, which depicts the mean energy spectrum
$\bar{\epsilon}_{n}=1/2\langle|u_{n}|^{2}\rangle$ of the deterministic and
noisy models. Note that shell models describe fluids effectively in $0$ space
dimensions, so that equipartition corresponds to a constant value
$(1/2)\varrho\langle|u_{n}|^{2}\rangle=k_{B}T,$ or
$1/2\langle|u_{n}|^{2}\rangle=\theta_{K}$ in dimensionless Kolmogorov units.
The deterministic model in the dissipation range displays stretched-
exponential decay

$\langle\epsilon_{n}\rangle\sim\exp(-c(k_{n}\ell_{K})^{\gamma}),\quad\gamma=\log_{2}\left(\frac{1+\sqrt{5}}{2}\right)$
(10)

which was verified by plotting local stretching exponents

$\gamma_{n}=\log_{2}\left|\ln\langle|u_{n+1}|^{2}\rangle\right|-\log_{2}\left|\ln\langle|u_{n}|^{2}\rangle\right|$
(11)

(see Inset of Fig. 2). Such decay is expected for deterministic shell models
Schörghofer  _et al._ (1995); L’vov  _et al._ (1998), but is not observed at
all once thermal noise is included. The stretched exponential decay is then
completely erased at shell-numbers above an equilibration scale $N_{e}=4$,
where the modal energy is close to the equipartition value $\theta_{K}$. This
equilibration wavenumber is just 16 times larger than the Kolmogorov
wavenumber. It should be emphasized that our shell model must underestimate
how rapidly the dissipation range is overtaken by thermal noise in comparison
to true hydrodynamic turbulence. Indeed, the latter has a faster exponential
spectral decay in the dissipation range Khurshid  _et al._ (2018) compared to
stretched-exponential for the shell model and, furthermore, the equipartition
spectrum $E(k)$ is rising in hydrodynamic turbulence $\propto k^{2}$ and is
decaying as $E(k_{n}):=\bar{\epsilon}_{n}/k_{n}\sim\theta_{K}/k_{n}$ in the
shell model.

In contrast to the dissipation range spectrum for the noisy model, the
inertial range energy spectrum shown in Fig. 2 is seemingly unchanged by
thermal noise, which is an indication that the deterministic model is adequate
for modeling fluctuations in the inertial range. The latter is known to
exhibit intermittency realized through bursts that form at large scales and
propagate to the dissipation range. Such bursts should be expected to cause
the equilibration shell-number $N_{e}$ to fluctuate strongly in time. To
investigate this issue, we defined an instantaneous equilibration shell-number
$N_{e}(u)$ as the smallest integer such that time-averages of $\epsilon_{n}$
over one Kolmogorov time are below $2\theta_{K}$ for all $n\geq N_{e}(u)$. We
illustrate this concept in Fig. 3, which plots the modal energy averaged over
one Kolmogorov time as a function of wavenumber, for one typical shell-model
realization and two extreme ones. We interpret the two extreme events as a
large burst pushing $N_{e}(u)$ to a high value and as a deep lull allowing
$N_{e}(u)$ to recede to a lower value. The underlying picture of intermittency
is accepted for turbulence both in Navier-Stokes equation and in shell models,
where it is attributed to self-similar solutions Daumont  _et al._ (2000);
Mailybaev (2013) that in the inviscid limit propagate in finite time to
infinite wavenumbers. The same mechanism underlies the fluctuating nature of
the “local viscous shellnumber” in the deterministic model

$N_{vis}(u):=\min\left\\{n:\,\frac{|u_{n}|}{\nu k_{n}}\leq 1\right\\},$ (12)

defined in analogy to the “local viscous/cutoff length” $\eta({\bf
x},t)\sim\frac{\nu}{\delta_{\eta}u({\bf x},t)}$ of hydrodynamic turbulence
Paladin and Vulpiani (1987); Schumacher (2007).

To characterize the statistical properties of the equilibration $N_{e}(u)$ and
viscous $N_{vis}(u)$ scales we plot the probability density function (PDF) of
their temporal fluctuations for the noisy Sabra model in Fig. 4. The PDF of
$N_{e}(u)$ is well approximated at the core of the distribution by a discrete
Gaussian $p(n)=e^{-(n-2)^{2}/2}/\Theta$ Agostini and Améndola (2019). This is
somewhat surprising given the obvious asymmetry between small and large
scales, but presumably such asymmetry is evidenced better in the extreme tails
of the PDF which are not well-resolved with our data from 300 turnover times.
Plotted as well in Fig. 4 is the PDF of $N_{vis}(u),$ defined as in (12). This
PDF is plotted together with a standard multifractal model of the type
developed for deterministic Navier-Stokes Biferale (2008), here constructed
using the ansatz Bowman  _et al._ (2006)

$N_{vis}(h)={\rm
Round}\left[\log_{2}(Re)\left(\frac{1}{1+h}-\frac{3}{4}\right)\right],$ (13)

where “Round” denotes rounding to the nearest integer, and the PDF of the
Hölder exponent $h$ is taken to be $P(h)\propto Re^{D(h)/(1+h)}$ for
multifractal dimension spectrum $D(h)=\min_{p}\\{ph-\zeta(p)\\}$ with
$\zeta(p)$ evaluated from our numerical simulation (see Fig. 5). The
multifractal model is in reasonable agreement with the PDF of $N_{visc}$.
Furthermore, for each realization that we used to calculate the PDF’s of
$N_{visc}(u)$ and $N_{e}(u)$ we calculated a shift $\Delta
N(u)=N_{visc}(u)-N_{e}(u)$ and its probability distribution in the steady-
state. This distribution is peaked at $\Delta n=3$ and indeed supports an
interpretation of the fluctuations of both $N_{visc}$ and $N_{e}$ being due to
bursting nature of the inertial range. A reasonable interpretation is that, in
each realization, the fluctuating viscous scale is followed by a stretched
exponential decay, which gets overtaken within just a couple of shells by
equipartition at the level of thermal energy $\theta_{K}$.

Figure 3: Modal energies of a lull (blue), a burst (red) and a typical
realization (green), averaged over 1 Kolmogorov time, plotted versus $n$.
Figure 4: PDF’s of $N_{vis}(u)$ (blue) and $N_{e}(u)$ (red), along with a
simple multifractal model for the PDF of $N_{vis}(u)$ (pale blue) and a fit by
a discrete Gaussian distribution for the PDF of $N_{e}(u)$ (pale red). Inset:
The PDF of the shift $\Delta N(u)$ in each realization. Standard errors of the
mean for $N_{vis}(u)$ and $N_{e}(u)$ are smaller than the marker size. Figure
5: Structure functions for the deterministic model (dashed lines), for the
noisy model (solid lines), and power-law fits in the inertial range (dotted
lines).

The fluctuating nature of the equilibration scale $N_{e}(u)$ has important
implications for the scale at which thermal effects appear in statistical
averages. We consider the velocity structure functions, which we define as

$\|u_{n}\|_{p}=\langle|u_{n}|^{p}\rangle^{1/p}$ (14)

with a $p$-th root. Here $\langle\cdot\rangle$ denotes the steady-state
average. Just as for the deterministic Sabra model L’vov  _et al._ (1998), the
noisy Sabra model exhibits power-law decay of structure functions with
wavenumber, $\|u_{n}\|_{p}\propto k_{n}^{-\sigma(p)}$, where
$\sigma(p)=\zeta(p)/p$ in terms of the standard exponents $\zeta(p)$, as shown
in Fig. 5. To the accuracy of our calculation, the corresponding exponents
$\sigma(p)$ for noisy and deterministic models are indistinguishable.

The inertial range intermittency is typically associated with nonlinear
dependence of scaling exponents $\sigma(p)$ on $p$. Looking at structure
functions with varying $p$ is also helpful for elucidation of the interplay of
intermittency and noise. Higher positive values of $p$ assign a higher weight
to events with larger value of $|u_{n}|$. In contrast negative values of $p$
shift weight to events with smaller values of $|u_{n}|$. This effect is
clearly seen in Fig. 5, where one can read off the cut-off shellnumber
$N_{e}(p)$ at which the $p$th structure-function attains its equipartition
value $[\Gamma(1+\frac{p}{2})]^{1/p}\theta_{K}$. Evidently $N_{e}(p)$ exhibits
strong $p$-dependence, with smaller $N_{e}(p)$ for negative values of $p$, and
larger values of $N_{e}(p)$ for large positive $p$. Significantly, for
negative values of $p$, thermal effects can be seen even at length scales
above $\eta,$ inside the traditional inertial range. This suggests that a
similar result should hold for the fluctuating Navier-Stokes equation, but
experimental observation will be extremely challenging since the small
magnitude of the thermal velocities, 3-4 orders of magnitude less than the
Kolmogorov velocity $u_{\eta},$ will require very high precision of velocity
measurements to be resolved.

Effect of spatial dimension. Up to now, we have ignored the potentially
profound effect of space dimensionality $d$ on the character of fluctuating
hydrodynamics. This effect can be simply understood by the estimate
$u_{\ell}^{th}\sim(k_{B}T/\rho\ell^{d})^{1/2}$ for the magnitude of a thermal
velocity fluctuation at length-scale $\ell$ in space-dimension $d$. It follows
that the Reynolds number of the fluctuating hydrodynamic equation (1) in the
thermal equipartition regime is

$Re^{th}_{\ell}:=\frac{u_{\ell}^{th}\ell}{\nu}=\left(\frac{k_{B}T}{\rho\nu^{2}\ell^{d-2}}\right)^{1/2}$

at length-scale $\ell$. Clearly for $d>2,$ $Re_{\ell}^{th}$ becomes increasing
large with decreasing scale $\ell$. For $d<2$ $Re_{\ell}^{th}$ becomes smaller
with decreasing $\ell$. This fact is well-known and can be obtained by more
sophisticated renormalization group analysis Forster  _et al._ (1976, 1977),
according to which thermal fluids are UV asymptotically free for $d<2$ but UV
strongly coupled for $d>2.$ These considerations apply to our shell model
which has effectively $d=0$ and, indeed, we observe in our simulations that
the noisy turbulent model becomes almost UV free at high-wavenumbers, with
individual shell modes $u_{n}$ governed to good approximation by independent
linear Langevin equations. Since 3D fluctuating hydrodynamics instead becomes
strongly coupled in the UV, it might be doubted that our shell model properly
represents the physics at scales where thermal fluctuations become important.
However, the justification for our shell model lies in the extremely small
values of $u_{\ell}^{th}$ for turbulent flows at scales $\ell\lesssim\eta.$ A
dimensionless measure of this smallness is $Re^{th}_{\eta}=\theta_{K}^{1/2},$
which is of order $10^{-4}$ in the atmospheric boundary layer. A general
calculation Eyink  _et al._ (2021) confirms that fluctuating hydrodynamics for
any molecular fluid in $d=3$ will remain weakly coupled until $\ell$ is
comparable to the size of a molecule! Thus, the strong-coupling regime in the
model at scales smaller than this is of no physical relevance.

Conclusion. In this Letter we provided theoretical order of magnitude
estimates, corroborated by numerical simulations of the Sabra shell model of
the turbulent cascade that thermal noise cannot be neglected in the
dissipation range of incompressible turbulence, in contrast to the naive
expectation that thermal effects are only relevant in molecular fluids at
scales comparable to the mean free path length Von Neumann (1963). Our
arguments and numerical results thus confirm and extend the prescient ideas of
Betchov Betchov (1957, 1961, 1964). At the moment there are almost no
experimental measurements below the Kolmogorov length to challenge the naive
expectation that thermal noise becomes important only at the scale of the mean
free path. However, there are many practically relevant processes that take
place at sub-Kolmogorov scales of turbulent flow and therefore there is an
urgent need for new experimental methods that can probe velocity fluctuations
at those small scales with high accuracy to check the predictions made here.
Perhaps the first evidence of thermal effects will be indirect, through the
competing effects of turbulent and thermal fluctuations on the various
physical phenomena strongly sensitive to sub-Kolmogorov turbulent scales. For
example, we expect to see the influence of thermal noise on the Batchelor
range of high Schmidt-number passive scalars, which extends well below the
Kolmogorov scale. Another implication of our results is that there should be a
possibly subtle influence of thermal noise on the inertial range of
turbulence, through sensitivity of the equations of motion to perturbations
Mailybaev (2016); Thalabard  _et al._ (2020).

This work was supported by the Simons Foundation through Grant number 662985
(NG) and Grant number 663054 (GE).

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# Minimal surfaces containing an epitrochoid as a geodesic

Shin Kaneda

(September 16, 2022)

###### Abstract

We show a complete minimal immersion cannot have a certain kind of epitrochoid
as a geodesic.

## Introduction

The Björling problem for minimal surfaces in $\mathbb{R}^{3}$ asks for the
existence of a minimal surface containing a given curve in $\mathbb{R}^{3}$
and a given unit normal coincides with its Gauss map. The representation
formula of the surface of this problem’s solution is given by H. A. Schwarz.
See for instance, [M], [LW], [MW], and so on.

We consider minimal surfaces containing planar curves as geodesics. For
example, a catenoid contains a circle and catenaries as geodesics (See Figure
1), Catalan surface contains a cycloid and a parabola (See Figure 2). The
minimal surfaces containing a conic curve as a geodesic are studied in [AGS],
[P], and so on. Note that the Catalan surface and the minimal surfaces
containing a conic curve as a geodesic are not immersed, while the catenoid is
immersed and complete.

---
Figure 1:  Catenoid and planar curves, a circle and catenaries, contained as
geodesics.
---
Figure 2:  Catalan surface and planar curves, a cycloid and a parabola,
contained as geodesics.

In [FS], Fujimori and Shoda constructed complete minimal immersions with
finite total curvature. These surfaces contain a planar curve as a geodesic.
These curves really look like epitrochoids (See Figure 3). So the author tried
to prove that these curves are indeed epitrochoids. Then, surprisingly, we
found that these curves are different from the epitrochoids. In fact, we prove
the following :

######  Main Theorem.

A complete minimal immersion in $\mathbb{R}^{3}$ cannot have a single-wrapped
epitrochoid as a geodesic.

For the definition of single-wrapped epitrochoid, see Definition 4. By this
theorem, we see that the surfaces constructed in [FS] do not contain an
epitrochoid as a geodesic because these surfaces are immersed and complete.

---
Figure 3: The minimal surface constructed in [FS] (left), a half cut away from
the surface by $x_{1}x_{2}$-plane (middle), and its intersection with the
$x_{1}x_{2}$-plane (right).

##  1 Preliminaries

######  Theorem 1 (The Weierstrass representation [AGS, Theorem 22.27]).

Let $(g,\eta)$ be a pair of a meromorphic function $g$ and a holomorphic
differential $\eta$ on a Riemann surface $M$ such that

$(1+|g|^{2})^{2}\eta\bar{\eta}$ (1)

gives a Riemannian metric on $M$. We set

$\Phi=\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}=\begin{pmatrix}(1-g^{2})\eta\\\ i(1+g^{2})\eta\\\
2g\eta\end{pmatrix}.$ (2)

Suppose that

$\operatorname{Re}\oint_{\gamma}\Phi={\bf{0}}$ (3)

holds for any $\gamma\in H_{1}(M,\mathbb{Z}).$ Then

$f(z)=\operatorname{Re}\int_{z_{0}}^{z}\Phi:M\to\mathbb{R}^{3},\qquad(z_{0}\in
M)$ (4)

defines a minimal immersion in $\mathbb{R}^{3}$.

The pair $(g,\eta)$ in Theorem 1 is called the Weierstrass data of $f$.

######  Remark 2.

Let $(g,\eta)$ be the Weierstrass data of $f$. Then $(g,\eta)$ can be written
as

$g=\frac{\phi_{3}}{\phi_{1}-i\phi_{2}},\qquad\eta=\phi_{1}-i\phi_{2}.$ (5)

Moreover, $g$ coincides with the composition of the unit normal $\nu$ of $f$
and the stereographic projection $\sigma$ from the north pole, that is,
$g=\sigma\circ\nu$. Thus we call $g$ the Gauss map of $f$.

######  Theorem 3 (Schwarz [AGS, Section 22.6]).

Let $c:I\to\mathbb{R}^{3}$ be a real analytic regular curve and
$n:I\to\mathbb{R}^{3}$ a real analytic vector field along $c$ such that

$\langle{n(t)},{n(t)}\rangle=1,\qquad\langle{n(t)},{c^{\prime}(t)}\rangle=0,\qquad
t\in I,$

where $I$ is an interval in $\mathbb{R}$. Then, for sufficiently small
$\varepsilon>0$ there exists a unique minimal immersion
$f:I\times(-\varepsilon,\varepsilon)\to\mathbb{R}^{3}$ satisfying

$f(t)=c(t),\qquad\nu(t)=n(t),\qquad t\in I,$

where $\nu$ is the unit normal of $f$. Moreover, $f$ can be written on
$I\times(-\varepsilon,\varepsilon)$ as

$f(z)=\operatorname{Re}\left\\{c(z)-i\int_{z_{0}}^{z}n(w)\times
c^{\prime}(w)dw\right\\},$ (6)

where $c(z),n(z)$ are extensions of $c(t),n(t)$ to functions of a complex
variable $z=t+is\in I\times(-\varepsilon,\varepsilon)$.

We consider minimal immersions containing a real analytic regular planar curve
that is not a straight line $c(t)=(x(t),y(t),0)$ as a geodesic. We also
suppose that $c(t)$ is regular, that is,
$x^{\prime}(t)^{2}+y^{\prime}(t)^{2}>0$ for any $t\in I$. Since
$c^{\prime\prime}(t)$ is contained in the $x_{1}x_{2}$-plane and $c$ is a
geodesic of $f$, the unit normal $n$ of $c$ can be written as

$n(t)=\frac{1}{\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}}(-y^{\prime}(t),x(t),0).$

Applying (6) to $c$ and $n$, we have

$f(z)=\operatorname{Re}\int_{z_{0}}^{z}\left(x^{\prime}(w),y^{\prime}(w),i\sqrt{x^{\prime}(w)^{2}+y^{\prime}(w)^{2}}\right)dw.$
(7)

Moreover, from (5), we have

$g=i\sqrt{\frac{x^{\prime}(z)+iy^{\prime}(z)}{x^{\prime}(z)-iy^{\prime}(z)}},\qquad\eta=(x^{\prime}(z)-iy^{\prime}(z))dz.$
(8)

Let $c(t)=(x(t),y(t),0)$ be an epitrochoid. Then it is written by

$\begin{cases}x(t)=(r_{c}+r_{m})\cos
t-\lambda\cos\left(\dfrac{r_{c}+r_{m}}{r_{m}}t\right),\vspace{4pt}\\\
y(t)=(r_{c}+r_{m})\sin
t-\lambda\sin\left(\dfrac{r_{c}+r_{m}}{r_{m}}t\right),\end{cases}$ (9)

where $r_{c}>0$ and $r_{m}>0$ are radii of a fixed circle and a rolling circle
around outside of a fixed circle respectively, $\lambda>0$ is a distance
between a center of a rolling circle and a point on the epitrochoid (See
Figure 4). For the curve $c(t)$ to be the regular, we assume that $\lambda\neq
r_{m}$.

$r_{c}$$r_{m}$$\lambda$

Figure 4: The thick curve indicates an epitrochoid, the big circle is a fixed
circle, and the small circle is a rolling circle.

Without loss of generality, we may assume that $r_{c}=1$.

######  Definition 4.

An epitrochoid (9) is called single-wrapped if there exists
$k\in\mathbb{Z}_{>0}$ such that $r_{m}=1/(k+1)$. Note that if an epitrochoid
is single-wrapped, then it is closed when the rolling circle wraps once around
the fixed circle.

We consider a single-wrapped epitrochoid, that is, we set
$r_{m}=1/(k+1),k\in\mathbb{Z}_{>0}$. Then (9) becomes

$\begin{cases}x(t)=\left(1+\dfrac{1}{k+1}\right)\cos
t-\lambda\cos\left(\dfrac{1+1/(k+1)}{1/(k+1)}t\right),\vspace{4pt}\\\
y(t)=\left(1+\dfrac{1}{k+1}\right)\sin
t-\lambda\sin\left(\dfrac{1+1/(k+1)}{1/(k+1)}t\right).\end{cases}$

---
Figure 5: Single-wrapped epitrochoids (left : $k=2$, middle : $k=3$, right :
$k=4$) (${\lambda}>r_{m}$)

Scaling up this curve, we can set

$\begin{cases}x(t)=\left(k+2\right)\cos
t-(k+1)\lambda\cos\left(k+2\right)t\vspace{4pt}\\\ y(t)=\left(k+2\right)\sin
t-(k+1)\lambda\sin\left(k+2\right)t.\end{cases}$ (10)

See Figure 5.

##  2 The proof of the main theorem

We consider a complete minimal immersion containing a single-wrapped
epitrochoid as a geodesic. Applying (7) and (10) , we have

$f(z)=\operatorname{Re}\int_{z_{0}}^{z}\begin{pmatrix}-(k+2)\sin
z+(k+1)(k+2)\lambda\sin(k+2)z\\\ (k+2)\cos z-(k+1)(k+2)\lambda\cos(k+2)z\\\
i(k+2)\sqrt{1+(k+1)^{2}\lambda^{2}-2\lambda(k+1)\cos(k+1)z}\end{pmatrix}dz.$
(11)

By (8),

$g=i\sqrt{\frac{e^{iz}-\lambda(k+1)e^{(k+2)iz}}{-e^{-iz}+\lambda(k+1)e^{-(k+2)iz}}},\qquad\eta=(k+2)(e^{-iz}+\lambda(k+1)e^{-(k+2)iz})dz.$

After the substitution $v=e^{iz}$ and rotating the surface in $\mathbb{R}^{3}$
by $-\pi/2$ about the $x_{3}$-axis, we have

$g=\sqrt{\frac{v^{k+3}(1-\lambda(k+1)v^{k+1})}{-v^{k+1}+\lambda(k+1)}},\qquad\eta=(k+2)\frac{v^{k+1}-\lambda(k+1)}{v^{k+3}}dv.$

We set

$w^{2}=\begin{cases}v(1-\lambda(k+1)v^{k+1})(\lambda(k+1)-v^{k+1})\qquad\text{
($k$ is even),}\\\ (1-\lambda(k+1)v^{k+1})(\lambda(k+1)-v^{k+1})\qquad\text{
($k$ is odd).}\end{cases}$

Case 1. $k$ is even.

Let $\overline{M}$ be the compact Riemann surface of genus $k+1$ defined by

$\overline{M}=\\{(v,w)\in(\mathbb{C}\cup\\{\infty\\})^{2}\>|\>w^{2}=v(1-\lambda(k+1)v^{k+1})(\lambda(k+1)-v^{k+1})\\},$

and set $M=\overline{M}\backslash\\{(0,0),(\infty,\infty)\\}.$ Then the
Weierstrass data of $f$ can be written as

$g=-\frac{wv^{\frac{k+2}{2}}}{v^{k+1}-\lambda(k+1)},\qquad\eta=\frac{v^{k+1}-\lambda(k+1)}{v^{k+1}}dv.$

It is easy to check that (1) degenerates at $v=\sqrt[k+1]{\lambda(k+1)}$ and
$v=\sqrt[k+1]{1/(\lambda(k+1))}$. Therefore $f$ is not an immersion at these
points. See Table 1.

$(v,w)$ | $(0,0)$ | $(\sqrt[k+1]{\lambda(k+1)},0)$ | $(\sqrt[k+1]{1/(\lambda(k+1))},0)$ | $(\infty,\infty)$
---|---|---|---|---
$g$ | $0^{k+3}$ | $\infty^{1}$ | $0^{1}$ | $\infty^{k+3}$
$\eta$ | $\infty^{2k+5}$ | $0^{3}$ | $0^{1}$ | $0^{1}$
Table 1: Orders of zeros and poles of $g$ and $\eta$
---
Figure 6:  The surface with $k=2$ (left), and a half cut away from the surface
by the $x_{1}x_{2}$-plane (right).

Case 2. $k$ is odd.

Let $\overline{M}$ be the compact Riemann surface of genus $k$ defined by

$\overline{M}=\\{(v,w)\in(\mathbb{C}\cup\\{\infty\\})^{2}\>|\>w^{2}=(1-\lambda(k+1)v^{k+1})(\lambda(k+1)-v^{k+1})\\},$

and set
$M=\overline{M}\backslash\\{(0,\pm{\sqrt{\lambda(k+1)}}),(\infty,\infty)\\}.$
Then the Weierstrass data of $f$ can be written as

$g=-\frac{wv^{\frac{k+3}{2}}}{v^{k+1}-\lambda(k+1)},\qquad\eta=\frac{v^{k+1}-\lambda(k+1)}{v^{k+1}}dz.$

It is easy to check that (1) degenerates at $v=\sqrt[k+1]{\lambda(k+1)}$ and
$v=\sqrt[k+1]{1/(\lambda(k+1))}$. Therefore $f$ is not an immersion at these
points. See Table 2.

$(v,w)$ | $(0,\pm{\sqrt{\lambda(k+1)}})$ | $(\sqrt[k+1]{\lambda(k+1)},0)$ | $(\sqrt[k+1]{1/(\lambda(k+1))},0)$ | $(\infty,\infty)$
---|---|---|---|---
$g$ | $0^{\frac{k+3}{2}}$ | $\infty^{1}$ | $0^{1}$ | $\infty^{\frac{k+3}{2}}$
$\eta$ | $\infty^{k+3}$ | $0^{3}$ | $0^{1}$ | $-$
Table 2: Orders of zeros and poles of $g$ and $\eta$
---
Figure 7: The surface with $k=3$ (left), and a half cut away from the surface
by the $x_{1}x_{2}$-plane (right).

## Acknowledgement

The author would like to thank the referee, and Professor Shoichi Fujimori and
Keisuke Teramoto for valuable comments and suggestions.

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Shin Kaneda
Department of Mathematics
Hiroshima University
Higashihiroshima, Hiroshima 739-8526, Japan
E-mail address<EMAIL_ADDRESS>