11institutetext: University College Dublin, Dublin, Ireland
11email: {susan.leavy,emilie.pine<EMAIL_ADDRESS>
# Industrial Memories: Exploring the Findings of Government Inquiries with
Neural Word Embedding and Machine Learning

Susan Leavy  Emilie Pine  Mark T Keane

###### Abstract

We present a text mining system to support the exploration of large volumes of
text detailing the findings of government inquiries. Despite their historical
significance and potential societal impact, key findings of inquiries are
often hidden within lengthy documents and remain inaccessible to the general
public. We transform the findings of the Irish government’s inquiry into
industrial schools and through the use of word embedding, text classification
and visualization, present an interactive web-based platform that enables the
exploration of the text to uncover new historical insights.

###### Keywords:

Word EmbeddingsText Classification Visualization Government Inquiry Reports

##  1 Introduction

The Irish Government published one of the most important documents in the
history of the state in 2009. It spanned over 1 million words and detailed the
findings of a 9 year investigation into abuse and neglect in Irish industrial
schools. The document however, remains largely unread. The structure and style
of the report obscures significant information and prevents the kind of
system-wide analysis of the industrial school regime which is crucial to
identifying recurring patterns and preventing the re-occurrence of such
atrocities.

The Industrial Memories project, publicly available
online111https://industrialmemories.ucd.ie, presents a web-based interactive
platform where the findings of CICA, commonly known as the Ryan Report, may be
explored by researchers and the general public in new ways. The machine
learning approach developed as part of this project addressed the challenge of
learning within the context of limited volumes of training data. Techniques
such as named entity recognition, word embedding and social network analysis
are also used to extract information and unearth new insights from the report.

##  2 System Overview

The Industrial Memories project developed a web-based platform where the
narrative form of the Ryan Report is deconstructed and key information
extracted. In the Ryan Report most of the chapters deal separately with
individual industrial schools. This segregation of information prevents a
system-wide analysis of common trends and patterns across the industrial
school system. To address this, a random forest classifier was employed to
annotate the report. An interactive web based platform enables users to
extract information and interact with the report using a comprehensive search
tool (Fig. 1). Word embeddings were also used to identify actions pertaining
to meetings and communication that occurred within the industrial school
system facilitating an analysis of the dynamics of power and influence using
social network analysis. Each individual named in the report was extracted
using the Stanford NE system [1] which for the first time, allows the history
of individuals to be traced over the entire industrial school system thus
responding to a call for increased open data in government [2].

Figure 1: Text Navigation Tool

###  2.1 Semantic Classification of Text

While some semantic categories or themes were conducive to extraction using a
rule based approach, categories including the transfer events, descriptions of
abuse and witness testimony were extracted using a random forest learning
algorithm. The random forest algorithm was chosen because, as an ensemble
learner that creates a ‘forest’ of decision trees by randomly sampling from
the training set, it is suited to learning from smaller datasets [4]. The Ryan
Report was represented in a relational database with annotated excerpts
comprising 6,839 paragraphs (597,651 words). Each paragraph was considered a
unit-of-analysis in the Report for the purposes of annotation.

###  2.2 Feature Extraction with Word Embedding

Feature selection was based on the compilation of domain-specific lexicons in
order to address the constraint in this project concerning low volumes of
available training data. As with many projects that are based in the
humanities, compiling training data is a manual process involving close
reading of texts. To address this, domain-specific lexicons were compiled
based on the entire dataset yielding high accuracy with low numbers of
training examples.

Lexicons were generated using the word2vec algorithm from a number of seed
terms [3]. Given that the word2vec algorithm performs best with larger
datasets than the Ryan Report, five ensembles were generated and the top
thirty synonyms for each seed term were extracted. Terms common across each
ensemble were identified to generate domain specific synonyms. Features for
the classification process were then derived from these lexicons.

Figure 2: Collocation Network of Entities in the Ryan Report

###  2.3 Exploring Power, Influence and Migration within the System with Word
Embedding and Network Analysis

Neural network analysis was used to generate lexicons describing interactions
in the form of meetings, letters and telephone calls to be compiled and
excerpts describing these communications extracted from the report. A social
network based on these excerpts combined with entities named in the report was
constructed along with a network detailing how entities were related in the
narrative of the Ryan Report through collocation within paragraphs (Fig. 2).
The sub-corpus that was generated by automatically extracting excerpts
detailing transfers was examined using association rule analysis revealing the
recurring patterns governing the movement of religious staff often in response
to allegations of abuse (Fig. 3).

Figure 3: System-Wide Analysis of the Transfer of Religious Throughout the
Industrial School System

##  3 Conclusion

The Industrial Memories project presents a state-of-the-art text mining
application to navigate the findings of the Irish Government’s Inquiry into
Child Abuse at Industrial Schools. A demonstration video of the system is
available online 222https://youtu.be/cV1xzuJ0dv0. Text analytic techniques
were used to uncover new insights and system-wide patterns that are
represented in an interactive web based platform accessible to the general
public.

## References

  * [1] Finkel, J.R., Grenager, T., Manning, C.: Incorporating non-local information into information extraction systems by gibbs sampling. In: Proceedings of the 43rd annual meeting on association for computational linguistics. pp. 363–370. Association for Computational Linguistics (2005)
  * [2] Kitchin, R.: The data revolution: Big data, open data, data infrastructures and their consequences. Sage (2014)
  * [3] Mikolov, T., Sutskever, I., Chen, K., Corrado, G.S., Dean, J.: Distributed representations of words and phrases and their compositionality. In: Advances in neural information processing systems. pp. 3111–3119 (2013)
  * [4] Polikar, R.: Ensemble based systems in decision making. IEEE Circuits and systems magazine 6(3), 21–45 (2006)


# A note on asymptotic density

Ryszard Frankiewicz and Joanna Jureczko111corresponding author

###### Abstract

It is proved that $P(\omega)/\triangle_{d}$, where $\triangle_{d}$ is the
ideal of sets of asymptotic density zero, is universal in the sense of
embeddings.

222Mathematics Subject Classification: 54B15, 03E20, 03E35, 03E40, 03E65.
Keywords: asymptotic density, Boolean algebra, embeddings, Cantor cube, Haar
measure, forcing, Cohen reals, limits, Hausdorff gaps, P-point, OK-point.

##  1 Introduction

We examine the algebra $P(\omega)/\triangle_{d}$, where $\triangle_{d}$ is the
ideal of sets of asymptotic density zero. We start this paper with Proposition
1, in which we show that $P(\omega)/fin$ is embedded into
$P(\omega)/\triangle_{d}$. Thus, one can suspect that
$P(\omega)/\triangle_{d}$ has a number of common features with the algebra
$P(\omega)/fin$. Actually, so, because, for example both algebras have no
$(\omega,\omega)$-gaps, (compare [6, p. 38] and Proposition 2 in this paper).
They are also different in some aspects, i.e., $P(\omega)/fin$ has no
$\omega$-limits, (see [6, p.33]) but, as we will show in Proposition 3,
$P(\omega)/\triangle_{d}$ has. Moreover, as we will show in Proposition 4,
$(P(\omega)/\triangle_{d})^{\omega}$ is isomorphic to
$P(\omega)/\triangle_{d}$, the same property is not true for $P(\omega)/fin$
even for the finite product, (see e.g.[7] p. 89 and 215 for details).

Furthermore, the space $\omega^{*}=st(P(\omega)/fin)$ was recognized as
homogeneous up to 1956 when W. Rudin proved that under CH it has $P$-points,
(see [13] and [6, p. 79]). In 1978 K. Kunen proved in ZFC that there exist
weak P-points in $\omega^{*}$, (see [9]), more precisely, he showed that in
$\omega^{*}$ there are weak $P$-points which are not $P$-points. (Later in
[14] there was proved that in some models $P$-points do not exist in
$\omega^{*}$). As we will show in Theorem 1, $P(\omega)/\triangle_{d}$ has a
weak $P$-point, (our argumentation is based on [8, 9], where there is
constructed a special kind of family, namely an independent linked family).

By well known Parovičenko theorem, (see [6, p.43-45]), $P(\omega)/fin$ is
universal in the sense of embeddings of algebras of cardinality continuum.
However, as was shown by Shelah in [14] the algebra $LM/\triangle$, (where
$LM$ is the collection of all Lebesgue measurable sets and $\triangle$ is the
ideal of sets of Lebesgue measure zero) consistently does not have to be
embedded in $P(\omega)/fin$. However, it is true that after adding
$\omega_{2}$ Cohen reals one can embed $LM/\triangle$ into $P(\omega)/fin$,
but it is not equivalent to CH, (see [2]). In the presence of Proposition 1,
we can conclude that there are such models for which there is no embedding
$P(\omega)/\triangle_{d}$ into $P(\omega)/fin$. Despite this result, we show
that after adding $\omega_{2}$ Cohen reals $P(\omega)/\triangle_{d}$ can be
embedded in $P(\omega)/fin$, (see Theorem 2 in which the proof we use the
ideas presented in [2]).

In [5] it is proved that there is a measure preserving embedding from
$LM/\triangle$ into $P(\omega)/\triangle_{d}$. This statement leads us to the
most spectacular result of this paper which we include in Theorem 3, i. e.
there exists an embedding $\phi\colon H(2^{\mathfrak{c}})/\triangle_{H}\to
P(\omega)/\triangle_{d}$ such that $d(\phi(a))=~{}\mu(a),$ $a\in
H(2^{\mathfrak{c}})/\triangle_{H}$, where $H(2^{\mathfrak{c}})$ is the family
of all Haar measurable sets of $2^{\mathfrak{c}}$, $\mu$ is the Haar measure
on the Cantor cube $2^{\mathfrak{c}}$ and $\triangle_{H}$ is the ideal of sets
of Haar measure zero. (In the proof of Theorem 3 we use only combinatorial
methods and ideas presented in paper [5]).

The paper is organized in two sections. In Section 2, there are gathered basic
definitions and notations used in the further part of this paper. Section 3 is
divided into four subsections containing: embeddings into
$P(\omega)/\triangle_{d}$, the existence of a weak $P$-point in
$st(P(\omega)/\triangle_{d})$, the embeddings $P(\omega)/\triangle_{d}$ into
$P(\omega)/fin$ (under assumptions mentioned above), the measure preserving
embedding $H(2^{\mathfrak{c}})/\triangle_{H}$ into $P(\omega)/\triangle_{d}$.

##  2 Definitions and previous results

1.1. An ordinal $\omega$ denotes the set of non-negative integers, $fin$
denotes the ideal of all finite subsets of $\omega$.

1.2. Let $A\subseteq\omega$. Lower asymptotic density $d_{*}(A)$ and upper
asymptotic density $d^{*}(A)$ is defined respectively

$d_{*}(A)=\lim_{n\to\infty}\inf\frac{|A\cap n|}{n}\ \textrm{ and }\
d^{*}(A)=\lim_{n\to\infty}\sup\frac{|A\cap n|}{n}.$

If $d_{*}(A)=d^{*}(A)=:d(A)$, then $d(A)$ is called (asymptotic) density. By
$\triangle_{d}$ we denote the ideal of subsets of (asymptotic) density zero.
Notice that sets with density do not form an algebra, but in many aspects they
behave as an algebra. In cases where density may not exist we will consider
the upper density.

1.3. Let $\mathbb{A}$ and $\mathbb{B}$ be Boolean algebras. We say that
$\phi\colon\mathbb{A}\to\mathbb{B}$ is a regular embedding iff
(I) $\phi$ is homomorphism,
(II) $\phi$ is embedding,
(III) if $\mathcal{A}$ is a maximal antichain in $\mathbb{A}$, then
$\\{\phi(A)\colon A\in\mathcal{A}\\}$ is a maximal antichain in $\mathbb{B}$,
(see also [12, vol. 1]).

1.4. Let $\mathbb{B}$ be a Boolean algebra. A gap (or cut) is a pair $\langle
S,T\rangle$, where $S,T\subseteq\mathbb{B}$ such that $a\cdot b=\mathbb{O}$
for each $a\in S$ and $b\in T$. The element $c$ fills (or separates) a gap
$\langle S,T\rangle$ if $a\leq c$ and $c\cdot b=\mathbb{O}$ for each $a\in S$
and $b\in T$. A pair of cardinals $(\kappa,\lambda)$, where $\kappa=|S|$ and
$\lambda=|T|$ is called a type of $\langle S,T\rangle$ if there is no
$c\in\mathbb{B}$ which separates $\langle S,T\rangle$. The system without
separating elements is called a $(\kappa,\lambda)$-gap.

1.5. A descending chain $\\{a_{\alpha}\colon\alpha<\gamma\\}$ in Boolean
algebra $\mathbb{B}$ by a $\gamma$-limit (a limit of length $\gamma$), if
there is no positive $a$ such that $a<a_{\alpha}$ for each $\alpha<\gamma$.

1.6. Let $\mathbb{B}$ be a Boolean algebra. The gap determined by $c$ over
$\mathbb{B}$ is a pair $\langle A,B\rangle$, where $A$ consists of all
elements of $\mathbb{B}$ with $a\leq c$ and $B$ consists of all elements of
$\mathbb{B}$ with $c\leq b$. $\langle A,B\rangle$ is countably generated if
$A$ is a countably generated as an ideal and $B$ is countably generated as a
filter (possibly improper).

1.7. The $st(\mathbb{A})$ denotes the Stone space of $\mathbb{A}$. Then
$\beta\omega=st(P(\omega))$ and
$\beta\omega\setminus\omega=st(P(\omega)/fin)=\omega^{*}$, (see [3, 11] for
details).

1.8. Let $X$ be a topological space. A point $p\in X$ is called a $P$-point
iff $p\in\textrm{int}(\bigcap_{n}U_{n})$, whenever $\\{U_{n}\colon
n\in\omega\\}$ is a countable family of neighbourhoods of $p$. A point $p$ is
called a weak $P$-point iff $p$ is not a limit point of any countable subset
of $X\setminus\\{p\\}$. Clearly, every $P$-point is a weak $P$-point, but not
conversely, (see [9]).

1.9. $LM$ denotes the family of all Lebesgue measurable sets in $2^{\omega}$,
$\lambda$ denotes the Lebesgue measure and $\triangle$ denotes the ideal of
subsets of Lebesgue measure zero.

1.10. $H(2^{\kappa})$ denotes the family of all Haar measurable sets of
$2^{\kappa}$, where $\kappa$ is an infinite cardinal. The Haar measure $\mu$
is generated by the basic sets $[s]=\\{f\in\\{0,1\\}^{\kappa}\colon s\subseteq
f\\}$, where $s\in\\{0,1\\}^{<\omega}$ and $\mu([s])=\frac{1}{2^{|s|}}$. The
ideal of sets of Haar measure zero is denoted by $\triangle_{H}$.

Notice that we can consider $2^{\mathfrak{c}}$ as the product space
$(2^{\omega})^{\mathfrak{c}}$. If we endow the Cantor cube $2^{\mathfrak{c}}$
with the structure of topological group with Haar measure $\mu$ then there is
a continuous map $\varphi\colon 2^{\mathfrak{c}}\to 2^{\omega}$ satisfying
$\varphi(\mu)=\lambda$, where $\lambda$ is the Lebesgue measure on
$2^{\omega}$, (it is folklore). Then all theorems concerning the base of such
a product and product measure are applied (see [1, 4]).

1.11. For definitions not cited here, we refer the reader to [1, 3, 4, 6, 7,
11, 12, 15].

##  3 Main results

### 3.1

###### Proposition 1

There exists a regular embedding $P(\omega)/fin$ into
$P(\omega)/\triangle_{d}$.

Proof. Let $\\{k_{n}\colon n\in\omega\\}$ be a sequence of elements of
$\omega$ such that $k_{n}<~{}k_{n+1},$ $n\in\omega$ and

$\lim_{n\to\infty}\frac{k_{n+1}-k_{n}}{k_{n+1}}=1.$

Let $f\colon P(\omega)/fin\to P(\omega)/\triangle_{d}$ be a function such that

$f([A])=[\bigcup_{n\in A}[k_{n},k_{n+1})],$

where $[A]\in P(\omega)/fin$ and $[\bigcup_{n\in A}[k_{n},k_{n+1})]]\in
P(\omega)/\triangle_{d}$. Clearly, $f$ is a homomorphism and even embedding,
because $d(\bigcup_{n\in A}[k_{n},k_{n+1}))>0$. If $\mathcal{A}$ is a maximal
antichain in $P(\omega)/fin$, then it is clear that $\\{f([A])\colon
A\in\mathcal{A}\\}$ is a maximal antichain in $P(\omega)/\triangle_{d}$.

###### Proposition 2

$P(\omega)/\triangle_{d}$ has no $(\omega,\omega)$-gaps.

Proof. Consider a gap in $P(\omega)/\triangle_{d}$

$\langle\\{[A_{n}]\colon n\in\omega\\},\\{[B_{m}]\colon m\in\omega\\}\rangle,$

such that
(i) $d^{*}(A_{n})>0,d^{*}(B_{n})>0$ for $n,m\in\omega$,
(ii) $A_{\alpha}\cap B_{\beta}=_{*}0$ for any $\alpha,\beta\in\omega$, i.e.
$|A_{\alpha}\cap B_{\beta}|<\omega$.
Then $d(A_{\alpha}\cap B_{\beta})=0$ for any $\alpha,\beta\in\omega$.

Now for each $n\in\omega$ take $C_{n}\subseteq\omega$ such that
$\\{A_{n}\setminus C_{n}\colon n\in\omega\\}$ is an increasing sequence. Since
property AP holds, (see [10]), there is $C\subseteq\omega$ such that
(iii) $d^{*}((A_{n}\setminus C_{n})\setminus C)=0$
(iv) $d^{*}(C)=\lim_{n\to\infty}d^{*}(A_{n}\setminus C_{n})$.
Then for each $n\in\omega$ we have

$d^{*}(C\cap B_{m})=d^{*}((A_{n}\setminus C)\cup(A_{n}\cap
B_{m}))=d^{*}(A_{n}\setminus C)+d^{*}(A_{n}\cap B_{m})=0.$

The proof is completed

###### Proposition 3

$P(\omega)/\triangle_{d}$ has an $\omega$-limit.

Proof. Consider a chain $\\{a_{n}\colon n\in\omega\\}$ such that
(i) $a_{n}\geqslant a_{n+1}$,
(ii) $a_{n}=[A_{n}]\in P(\omega)/\triangle_{d}$,
(iii) $d(A_{n})=\frac{1}{2^{n}}$.
Observe that $A_{n}\supseteq A_{n+1}$ and $\bigcap_{n\in\omega}A_{n}=0$. Then
$d(\bigcap_{n\in\omega}A_{n})=0$ and $[\bigcap_{n\in\omega}A_{n}]=\mathbb{O}$.

###### Proposition 4

There exists an embedding $f\colon(P(\omega)/\triangle_{d})^{\omega}\to
P(\omega)/\triangle_{d}$.

Proof. Consider a sequence $\\{k_{n}\colon n\in\omega\\}$ of elements of
$\omega$ such that $k_{n}<~{}k_{n+1},$ $n\in\omega$ and

$\lim_{n\to\infty}\frac{k_{n+1}-k_{n}}{k_{n+1}}=1.$

Consider a partition of $\omega$ into infinite sets of positive density
$\\{A_{n}\colon n\in\omega\\}$ such that $k_{n}\in A_{n}$ and $k_{m}\not\in
A_{n}$ for any distinct $m,n\in\omega$. Hence $[A_{n}]\cap[A_{m}]=\emptyset$,
thus the family $\\{P(A_{n})/\triangle_{d}\colon n\in\omega\\}$ is a partition
of $P(\omega)/\triangle_{d}$. Obviously the mapping

$h_{n}\colon P(\omega)/\triangle_{d}\to P(A_{n})/\triangle_{d}^{n},$

where $\triangle^{n}_{d}$ is the ideal of subsets of $A_{n}$ of density zero,
is an isomorphism, because $A_{n}$ is infinite for any $n\in\omega$. Then

$(h_{n})_{n\in\omega}=h\colon(P(\omega)/\triangle_{d})^{\omega}\to\Pi_{n\in\omega}P(A_{n})/\triangle_{d}^{n}$

is also an isomorphism.

Consider a mapping

$g\colon P(\omega)/\triangle_{d}\to\Pi_{n\in\omega}P(A_{n})/\triangle_{d}^{n}$

such that $g(x)=(x\cdot a_{n})_{n\in\omega}$, where $a_{n}\in
P(A_{n})/\triangle_{d}^{n}$. Notice that $g$ is homomorphism because
$h_{n}=\varphi\circ g$, where $\varphi:(P(\omega)/\triangle_{d})^{\omega}\to
P(\omega)/\triangle_{d}$ is a projection. The mapping $g$ is also one-to-one
because if $x\not=\mathbb{O}$ in $P(\omega)/\triangle_{d}$, then $x\cdot
a_{n}\not=\mathbb{O}$ for some $n\in\omega$. Now take
$b=(b_{n})_{n\in\omega}\in\Pi_{n\in\omega}P(A_{n})/\triangle_{d}^{n}$ and
$x=\sum_{n\in\omega}b_{n}$. Then $g(x)=b$, because
$\\{P(A_{n})/\triangle_{d}\colon n\in\omega\\}$ is a partition. Hence $g$ is
an isomorphism. Then

$f=g^{-1}\circ h\colon(P(\omega)/\triangle_{d})^{\omega}\to
P(\omega)/\triangle_{d}$

is the required embedding.

### 3.2

###### Lemma 1

For any family $\mathcal{A}=\\{A_{n}\subseteq\omega\colon n\in\omega\\}$ and
$T\subset\omega$ be a set of indices such that $|T|<\omega$

$d^{*}(\bigcup_{n\in T}A_{n})\leq\sum_{n\in T}d^{*}(A_{n}).$

Moreover if $\Sigma_{n\in\omega}d^{*}(A_{n})<\infty$, then

$d^{*}(\bigcup_{n\in\omega}A_{n})\leq\sum_{n\in\omega}d^{*}(A_{n}).$

Proof. Enumerate $T=\\{0,...,m\\}$ for some natural number $m$ and take
$\\{A_{n}\subseteq\omega\in n\in T\\}$. Then by standard argumentation we have

$\frac{|(\bigcup_{m\in T}A_{m})\cap n|}{n}=\frac{|(A_{0}\cap
n)\cup...\cup(A_{m}\cap n)|}{n}\leq\frac{|A_{0}\cap
n|}{n}+...+\frac{|A_{m}\cap n|}{n}.$

Taking the limit supremum under $n\to\infty$ we obtain the first part of the
lemma.

Assuming $\Sigma_{n\in\omega}d^{*}(A_{n})<\infty$ and using standard
analytical arguments, we obtain the second part of the lemma.

###### Lemma 2

Let $\\{A_{n}\subseteq\omega\colon d^{*}(A_{n})\leqslant\frac{1}{n+1}\\}$ and
let $S\subset\omega$ be a set of indices such that $|S|<\omega$. Then
$d^{*}(\bigcup_{n\in S}A_{n})\leqslant 1$ and $\\{A_{n}\colon n\in S\\}$ is
$|S|-$centered, i.e. $\bigcap_{n\in S}A_{n}\not=\emptyset$. Moreover if
$\Sigma_{n\in\omega}d^{*}(A_{n})<\infty$, then
$d^{*}(\bigcup_{n\in\omega}A_{n})<1$ .

Proof. Assume that $|S|=m+1$ for some $m\in\omega$. By Lemma 1

$d^{*}(\bigcup_{n\in S}A_{n})\leq\sum_{n\in
S}d^{*}(A_{n})\leqslant(m+1)\cdot\frac{1}{m+1}=1.$

To verify that $\\{A_{n}\colon n\in S\\}$ is $|S|$-centered notice that
$d^{*}(\omega)=1$ and use the argumentation above.

For the second part of the lemma, notice that
$\sum_{n\in\omega}\frac{1}{2^{n}}$ is the geometric series, use the inequality
above, the fact that $d^{*}(\omega)=1$ and the standard analytical arguments.

###### Lemma 3

For each ultrafilter $\mathcal{U}\in st(P(\omega)/\triangle_{d})$ and for each
$\varepsilon>0$ there exists $A\in\mathcal{U}$ such that
$d^{*}(A)<\varepsilon$.

Proof. We start with $d^{*}(\omega)=1.$ Divide $\omega$ into two sets
$A_{0},A_{1}$ in this way that $d^{*}(A_{0})=d^{*}(A_{1})=\frac{1}{2}$ and
then $A_{0}$ divide into two sets $A_{00},A_{01}$ in this way that
$d^{*}(A_{00})=d^{*}(A_{01})=\frac{1}{2^{2}}.$ In the similar way divide
$A_{1}$ obtaining $d^{*}(A_{10})=d^{*}(A_{11})=\frac{1}{2^{2}}$. Continue this
procedure up to $f\in{{}^{k}}\\{0,1\\},$ for some $k\in\omega$. Let

$\mathcal{A}_{k}=\\{A_{f}\subseteq\omega\colon
d^{*}(A_{f})=\frac{1}{2^{k}},f\in{{{}^{k}}\\{0,1\\}}\\}.$

Then divide each $A_{f}\in\mathcal{A}_{k}$ into two sets
$A_{f^{\frown}0},A_{f^{\frown}1}$ in this way that
$d^{*}(A_{f^{\frown}0})=d^{*}(A_{f^{\frown}1})=\frac{1}{2^{k+1}}$, where
$f^{\frown}0,f^{\frown}1\in{{}^{k+1}\\{0,1\\}}$. Since $k$ is arbitrary, the
proof is complete.

###### Lemma 4

Let $\\{A_{n}\colon n\in\omega\\}$ be a family such that
$\Sigma_{n\in\omega}d^{*}(A_{n})\leq 1$. Then for every utrafilter
$\mathcal{U}\in st(P(\omega)/\triangle_{d})$ there exists an element
$B\in\mathcal{U}$ such that
$\bigcup_{n\in\omega}A_{n}\cap~{}B\not=~{}\emptyset$ and
$d^{*}(\bigcup_{n\in\omega}A_{n}\cap B)<1$.

Proof. The case when $\sum_{n\in\omega}d^{*}(A_{n})=1$ is obvious. Let
$\sum_{n\in\omega}d^{*}(A_{n})<1$. By Lemma 1, we have
$d^{*}(\bigcup_{n\in\omega}A_{n})<1.$ Then
$0<d^{*}(\omega\setminus\bigcup_{n\in\omega}A_{n})\leqslant 1.$ For each
$n\in\omega$ take

$\mathcal{B}_{n}=\\{B\in\mathcal{U}\colon A_{n}\cap B\not=\emptyset\\}.$

For completeness the proof it is enough to show that there exists $n_{0}$ such
that $\mathcal{B}_{n_{0}}\not=\emptyset$. If $\mathcal{B}_{n}=\emptyset$ for
all $n\in\omega$ then all elements of $\mathcal{U}$ would be in
$\omega\setminus\bigcup_{n\in\omega}A_{n}$. But $d^{*}(\bigcup\mathcal{U})=1$.
A contradiction.

###### Theorem 1

In $st(P(\omega)/\triangle_{d})$ there exists a weak P-point.

Proof. Consider a family

$\mathcal{A}=\\{A^{\prime}_{n}\subset\omega\colon
d^{*}(A^{\prime}_{n})=\frac{1}{2^{n+1}},n\in\omega\\}.$

Such a family is infinite by Lemma 3. For each $n\in\omega$ consider a family

$\mathcal{A}_{n}=\\{A^{n}=A^{\prime}_{n}\setminus B\colon
B\subseteq\omega\textrm{ and }d^{*}(B)<\frac{1}{2^{n+1}}\\}.$

Each $\mathcal{A}_{n}$ has cardinality continuum. Order each
$\mathcal{A}_{n}$, i.e.

$\mathcal{A}_{n}=\\{A^{n}_{\alpha}\colon\alpha<2^{\omega}\\}$

and construct the family

$\mathcal{C}_{0}=\\{\bigcup_{n\in\omega}A^{n}_{\alpha_{n}}\colon(\alpha_{n})\in[2^{\omega}]^{\omega}\\}.$

Since the construction above and Lemma 2, $\mathcal{C}_{0}$ is a filter which
contains only infinite subsets of $\omega$. By Lemma 4, $\mathcal{C}_{0}$ is
not a required weak P-point. Notice that in $st(P(\omega)/\triangle_{d})$
there are $2^{\mathfrak{c}}$ ultrafilters but they are generated by
$2^{\omega}$ subsets of $\omega$. Enumerate all subsets of $\omega$ by
$\mathcal{V}=\\{V_{\alpha}\colon\alpha<2^{\omega}\\}.$ By Lemma 3, for each
$V\in\mathcal{V}$ there exists a countable decreasing sequence

$V=V^{0}\supseteq V^{1}\supseteq V^{2}\supseteq...$

such $d^{*}(V^{i})\geq d^{*}(V^{j})$ for $i>j$ and
$\lim_{n\to\infty}d^{*}(V^{n})=0$. Thus we have to construct an increasing
sequence of filters $\\{\mathcal{C}_{\alpha}\in
P(\omega)/\triangle_{d}\colon\alpha<2^{\omega}\\}$ with the following
properties:
(i) $\forall_{V\in\mathcal{V}}\exists_{\alpha<2^{\omega}}\
V\in\mathcal{C}_{\alpha}\textrm{ or }\omega\setminus
V\in\mathcal{C}_{\alpha}$,
(ii)
$\forall_{V\in\mathcal{V}}\exists_{B\in\bigcup\mathcal{C}_{\alpha}}\exists_{V^{\prime}\subseteq
V}\ (d^{*}(V^{\prime})\leq d^{*}(V)\Rightarrow V^{\prime}\cap B=\emptyset)$.
Then $\mathcal{C}=\bigcup_{\alpha\in 2^{\omega}}\mathcal{C}_{\alpha}$ will be
the required ultrafilter.

We will proceed by transfinite induction in $2^{\omega}$ stages. Let

$\\{D^{\lambda}_{\mu,n}\colon\mu,\lambda<2^{\omega},1\leq n<\omega\\}$

be an independent linked family mod $fin$, see [9]. We will construct an
increasing sequence of filters
$\\{\mathcal{C}_{\alpha}\colon\alpha<2^{\omega}\\}$ of properties (i)-(ii) and
a decreasing sequence of indices $\\{M_{\alpha}\colon\alpha<2^{\omega}\\}$ of
properties
(iii) $|M_{\alpha}\setminus M_{\alpha+1}|<\omega$
(iv) $\\{D^{\lambda}_{\mu,n}\colon\mu,\lambda<2^{\omega},1\leq
n<\omega,\lambda\in M_{\alpha}\\}$ is independent linked family mod
$\mathcal{C}_{\alpha}$.

We start with the filter $\mathcal{C}_{0}$ constructed at the beginning of
this proof and $M_{0}=2^{\omega}$. For nonzero limit ordinals $\alpha$ we
define $\mathcal{C}_{\alpha}=\bigcup_{\beta<\alpha}\mathcal{C}_{\beta}$ and
$M_{\alpha}=\bigcap_{\beta<\alpha}M_{\beta}$. Now we will construct
$\mathcal{C}_{\alpha+1}$ and $M_{\alpha+1}$. To do this, consider three cases.

Case 1) $V_{\alpha}\not\in\mathcal{C}_{\alpha}$. Then there exist
$C\in\mathcal{C}_{\alpha}$, $0<d^{*}(C)<1$, $S\in[M_{\alpha}]^{<\omega}$,
$n_{\lambda}\in\omega,\ \sigma_{\lambda}\in[2^{\omega}]^{n_{\lambda}}$ such
that

$V_{\alpha}\cap C\cap\bigcap_{\lambda\in
S}(\bigcap_{\mu\in\sigma_{\lambda}}D^{\lambda}_{\mu,n_{\lambda}})=\emptyset.$

Then we take $M_{\alpha+1}=M_{\alpha}\setminus S$ and
$\mathcal{C}_{\alpha+1}=\langle\mathcal{C}_{\alpha},\bigcap_{\lambda\in
S}(\bigcap_{\mu\in\sigma_{\lambda}}D^{\lambda}_{\mu,n_{\lambda}})\rangle$,
(where $C=\langle A,B\rangle$ means that $C$ is generated by $A$ and $B$).
Then obviously $\omega\setminus V_{\alpha}\in\mathcal{C}_{\alpha+1}$.

Case 2) $V_{\alpha}\in\mathcal{C}_{\alpha}$ and there exists $n_{0}\in\omega$
such that $V^{n_{0}}_{\alpha}\not\in\mathcal{C}_{\alpha}$. Then there exist
$C\in\mathcal{C}_{\alpha}$, $0<d^{*}(C)<1$,
$S\in[M_{\alpha}]^{<\omega},n_{\lambda}\in\omega,\sigma_{\lambda}\in[2^{\omega}]^{n_{\lambda}}$
such that

$V^{n_{0}}_{\alpha}\cap C\cap\bigcap_{\lambda\in
S}(\bigcap_{\mu\in\sigma_{\lambda}}D^{\lambda}_{\mu,n_{\lambda}})=\emptyset.$

Then we take $M_{\alpha+1}=M_{\alpha}\setminus S$ and
$\mathcal{C}_{\alpha+1}=\langle\mathcal{C}_{\alpha},\bigcap_{\lambda\in
S}(\bigcap_{\mu\in\sigma_{\lambda}}D^{\lambda}_{\mu,n_{\lambda}})\rangle$.
Then obviously $\omega\setminus V^{n_{0}}_{\alpha}\in\mathcal{C}_{\alpha+1}$.

Case 3) $V_{\alpha}\in\mathcal{C}_{\alpha}$ and
$V^{n}_{\alpha}\in\mathcal{C}_{\alpha}$ for all $n\in\omega$. For each
$n\in\omega$ take $W^{n}_{\alpha}=\omega\setminus V^{n}_{\alpha}$.

Case 3a) there exists $n_{0}\in\omega$ such that
$W^{n_{0}}_{\alpha}\not\in\mathcal{C}_{\alpha}$. Consider a cut
$((W^{n}_{\alpha}\cap W^{n_{0}}_{\alpha}\colon
n\in\omega),(V^{n}_{\alpha}\colon n\in\omega))$. By Proposition 2 there exists
$B_{\alpha}\subseteq\omega$ such that $W^{n}_{\alpha}\cap
W^{n_{0}}_{\alpha}\subseteq^{*}B_{\alpha}$ and $V^{n}_{\alpha}\cap
B_{\alpha}=^{*}\emptyset$. Then we take
$M_{\alpha+1}=M_{\alpha}\setminus\\{\lambda_{n_{0}}\\}$ and
$\mathcal{C}_{\alpha}=\langle\mathcal{C}_{\alpha},B_{\alpha}\cup\bigcup_{n<n_{0}}(D^{\lambda_{n_{0}}}_{\alpha,n}\cap
W^{n_{0}}_{\alpha})\rangle$.

Case 3b) $V^{n}_{\alpha}\in\mathcal{C}_{\alpha}$ and there exists
$B^{n}\subseteq\omega$ such that $W^{n}_{\alpha}\cap
B^{n}\in\mathcal{C}_{\alpha}$ for all $n\in\omega$. Then by Lemma 3, the
construction of $\mathcal{C}_{0}$ and previous steps there exists
$n_{0}\in\omega$ such that $W^{N_{0}}_{\alpha}\cap
B^{n_{0}}\in\mathcal{C}_{\alpha}$ and $W^{n_{0}}_{\alpha}\cap(\omega\setminus
B^{n_{0}})\not\in\mathcal{C}_{\alpha}$. Now take
$\hat{V}^{n_{0}}_{\alpha}=V^{n_{0}}_{\alpha}\cup W^{N_{0}}_{\alpha}\cap
B^{n_{0}}\in\mathcal{C}_{\alpha}$ and
$\hat{W}=\omega\setminus\hat{V}^{n_{0}}_{\omega}\not\in\mathcal{C}_{\alpha}$.
Consider a cut

$((\hat{V}^{n_{0}}_{\alpha}\cup V^{n}_{\alpha}\colon
n\in\omega),(\omega\setminus(\hat{V}^{n_{0}}_{\alpha}\cup
V^{n}_{\alpha})\colon n\in\omega)).$

By Proposition 2 there exists $B_{\alpha}\subseteq\omega$ such that
$\omega\setminus(\hat{V}^{n_{0}}_{\alpha}\cup
V^{n}_{\alpha})\subseteq^{*}B_{\alpha}$ and $\hat{V}^{n_{0}}_{\alpha}\cup
V^{n}_{\alpha}=^{*}\emptyset$. Then we take
$M_{\alpha+1}=M_{\alpha}\setminus\\{\lambda_{n_{0}}\\}$ and

$\mathcal{C}_{\alpha}=\langle\mathcal{C}_{\alpha},B_{\alpha}\cup\bigcup_{n<n_{0}}(D^{\lambda_{n_{0}}}_{\alpha,n}\cap(\hat{V}^{n_{0}}_{\alpha}\cup
V^{n_{0}}_{\alpha}))\rangle.$

The proof is complete.

###### Corollary 1

$st(P(\omega)/fin)$ has a weak P-point.

Proof. Using the properties of (asymptotic) density and the argumentation used
in the previous proof, we obtain our claim.

### 3.3

###### Lemma 5

If $P$ is a partial ordering for adding $\kappa$ Cohen reals for some cardinal
$\kappa$, then in $V^{P}$ the gap determined by any element of
$P(\omega)/\triangle_{d}$ over $(P(\omega)/\triangle_{d})^{V}$ is countably
generated.

Proof. In the first part of the proof we consider adding one Cohen real. Let
$V^{P}=V[G]$, where $G\subseteq P$ and $G$ is a $V$-generic filter. Let
$c~{}\in P(\omega)/\triangle_{d}$. For each $g\in G$ let

$c_{g}=\min\\{a\in(P(\omega)/\triangle_{d})^{V}\colon g\Vdash
a\wedge\bar{c}=0\\},$

where $\bar{c}$ is the name for c. Thus $\\{c_{g}\colon g\in G\\}$ is the
first component of the gap. The second one is obtained similarly.

Since in $V[G]$ every element of $P(\omega)/\triangle_{d}$ is in $V[x]$ for
some Cohen real $x$. Now, we use the first part of the proof for each $c\in
P(\omega)/\triangle_{d}$. Then the proof is complete.

###### Lemma 6

Let $\mathbb{B}$ be a subalgebra of $P(\omega)/\triangle_{d}$ such that for
each $c\in P(\omega)/\triangle_{d}$ the gap determined by $c$ over
$\mathbb{B}$ is countably generated. Let $\mathbb{C}$ is the subalgebra
generated by $\mathbb{B}$ with some countable set $C\subseteq\omega$. Then the
gap determined by each $c\in P(\omega)/\triangle_{d}$ over $\mathbb{C}$ is
countably generated.

Proof. Let $\langle A,D\rangle$ be a gap such that

$A=\\{a\in\mathbb{B}\colon a\leqslant b\\}\textrm{ and
}D=\\{d\in\mathbb{B}\colon b\leqslant d\\}.$

Let $c\in\mathbb{C}$. Consider a gap $\langle A^{\prime},D^{\prime}\rangle$
such that

$A^{\prime}=\\{a^{\prime}\in\mathbb{B}\colon a^{\prime}\leqslant
b\vee(-c)\\}\textrm{ and }D^{\prime}=\\{d^{\prime}\in\mathbb{B}\colon
b\vee(-c)\leqslant d\\}.$

Take an arbitrary $a^{\prime}\in A^{\prime}$. Observe that $A$ is generated by
all $a^{\prime}\wedge c$. Since $A^{\prime}$ is countably generated, hence $A$
is also. Similarly, $D$ is countably generated because $D^{\prime}$ is
countably generated.

###### Lemma 7

(CH) If $P$ is the partial ordering for adding $\omega_{2}$ Cohen reals, then
there is enumeration of $P(\omega)/\triangle_{d}$ in $V^{P}$ of length
$\omega_{2}$ such that the gap determined by each element over the subalgebra
generated by the previous elements is countably generated.

Proof. Consider the algebra $\mathbb{A}_{\beta}$ in the extension of $V$ by
first $\beta$ Cohen reals, $|\mathbb{A}_{\beta}|=\omega_{1}$ for
$\beta<\omega_{2}$.

Let $\\{a_{\alpha}\colon\alpha\in\omega_{2}\\}$ be a sequence enumerating
$P(\omega)/\triangle_{d}$ such that

$\mathbb{A}_{\beta}=\\{a_{\alpha}\colon\omega_{1}\beta\leq\alpha<\omega_{1}(\beta+1)\\}.$

Let $\mathbb{A}$ be the algebra generated by
$\\{a_{\gamma}\colon\gamma<\beta<\omega_{2}\\}$. Let $\mathbb{A}_{\gamma}$ be
the algebra generated by $a_{\gamma}$, with $\gamma<\omega_{1}\beta$, where
$\alpha=\omega_{1}\beta+\eta$ for $\beta<\omega_{2}$ and $\eta<\omega_{1}$.

By Lemma 6, we only need to show that for any $a\in P(\omega)/\triangle_{d}$
the gap determined by $a$ over $\mathbb{A}$ is countably generated. The case
$\beta=0$ is clear. By Lemma 5, each element of $P(\omega)/\triangle_{d}$
determines a gap over $\mathbb{A}_{\eta}$ and is countably generated for any
$\eta$.

If $cf(\beta)=\omega_{1}$ or $\beta=\delta+1$ for some $\delta<\omega_{2}$,
then $\mathbb{A}$ is $\mathbb{A}_{\beta}$ or $\mathbb{A}_{\delta}$
respectively.

If $cf(\beta)=\omega$, then
$\mathbb{A}=\bigcup\\{\mathbb{A}_{\xi}\colon\xi<\beta\\}$ and the gap
determined by $a$ over $\mathbb{A}$ is the union of gaps determined by $a$
over $A_{\xi}$, $\xi<\beta$, each countably generated. The proof is complete.

###### Theorem 2

Assume (CH). After adding $\omega_{2}$ Cohen reals the natural embedding from
$P(\omega)/\triangle_{d}$ from the ground model can be extended to embedding
into $P(\omega)/fin$.

Proof. By Lemma 7, we can enumerate $P(\omega)/\triangle_{d}$ by
$\\{a_{\beta}\colon\beta\in\omega_{2}\\}$ in this way that the gap determined
by each $a_{\beta}$ in the subalgebra generated by $a_{\alpha},\alpha<\beta$
is countably generated.

By induction on $\beta$ for each algebra $\mathbb{A}_{\beta}$ generated by
$a_{\alpha},\alpha<\beta$ a continuous tower of embeddings $g_{\beta}$ of
$\mathbb{A}_{\beta}$ into $P(\omega)/fin$ will be defined.

We will use a partial order, defined by Kunen, for filling a gap in
$P(\omega)/fin$, (see [2]). Let $\langle A,B\rangle$ be a gap in
$P(\omega)/fin$. Let $P(A,B)$ be the partial ordering constructing of all
partial functions $p\in{{}^{\omega}2}$ such that $\\{n\colon p(n)=1\\}/fin$ is
in the ideal generated by $A$ and $\\{n\colon p(n)=0\\}/fin$ is in the filter
generated by $B$. Observe that if $\langle A,B\rangle$ is countably generated
then $P(A,B)$ has a countable dense set and a Cohen real will add a generic
filter for $P(A,B)$. If $C\subseteq\omega$ is added by $P(A,B)$ then $C/fin$
fills the gap $\langle A,B\rangle$.

Notice that there is a collection $\mathcal{A}$ of at most $\omega_{1}$ dense
subsets of $P(A,B)$ such that any filter over $P(A,B)$ which meets all dense
sets in $\mathcal{A}$ determines a subset $A\subseteq\omega$ such that
$g_{\beta+1}(a_{\beta})=A/fin.$ Since $A$ and $B$ are countably generated,
$P(A,B)$ has a countable dense set, so either $P(A,B)$ has an atom or it is
equivalent to the forcing of adding a Cohen real. Notice that there is a
filter meeting any collection of at most $\omega_{1}$ dense sets in the
partial ordering by adding a Cohen real. Hence, there exists a filter meeting
each element of $\mathcal{A}$. We leave to the reader the modifications needed
to construct an embedding extending a given ground model embedding.

### 3.4

###### Lemma 8

The Cantor cube $2^{\mathfrak{c}}$ contains a countable dense subset.

Proof. Consider a family $\mathcal{S}$ of all finite subsets of $\omega$ For
each $S\in\mathcal{S}$ consider a function $p_{S}\colon\omega\to\\{0,1\\}$
such that $p_{S}$ is constant on $S$ and constant on $\omega\setminus S$. The
family $\\{p_{S}\colon S\in\mathcal{S}\\}$ is countable.

We can consider $\bigcup\\{2^{n}\colon n\in\omega\\}$ instead of $2^{\omega}$.
Consider a family of functions $f_{S}\colon 2^{\omega}\to\\{0,1\\}$ such that
$f_{S}$ is constant on a finite $0-1$ sequence of $2^{\omega}$ determined by
$p_{S}$ and constant on its completion. The family

$D=\\{f_{S}\colon S\in\mathcal{S}\\}$

is countable. By Hewitt-Marczewski-Pondiczery Theorem (see, e. g. [4]), the
set $D$ is dense in $2^{2^{\omega}}$.

###### Lemma 9

In $P(\omega)/\triangle_{d}$ there is an independent family
$\\{(A^{0}_{\alpha},A^{1}_{\alpha})\colon\alpha\in 2^{\omega}\\}$ such that
(I) $d(A^{\varepsilon}_{\alpha})=\frac{1}{2}$ for each
$\varepsilon\in\\{0,1\\}$ and $\alpha\in 2^{\omega}$
(II) $\bigcap_{s}A^{\varepsilon}_{s}=\frac{1}{2^{|s|}}$ for each
$\varepsilon\in\\{0,1\\}$ and $s\in[2^{\omega}]^{<\omega}$.

Proof. By Lemma 8, the Cantor cube $2^{\mathfrak{c}}$ contains a countable
dense set $D$. Enumerate $D=\\{d_{n}\colon n\in\omega\\}$. By Lemma 2 and
Lemma 3 (particularly the proof of Lemma 3), we can obtain a partition of
$\omega$ into infinite sets indexed by element of $D$ such that
$d^{*}(B_{d_{n}})\leqslant\frac{1}{2^{n}}$ such that
$B_{d_{n+1}}\subseteq\omega\setminus\bigcup_{k\leqslant n}B_{d_{k}}$. Thus,
the mapping $D\mapsto\\{B_{d_{n}}\colon n\in D\\}$ is continuous function.
Since $\bigcup_{d_{n}\in D}B_{d_{n}}=\omega$ and by [4, Theorem 3.6.1 and
Corollary 3.6.12, p. 222] we can extend $g\colon\bigcup_{d_{n}\in
D}B_{d_{n}}\to D$, (in fact the mapping $\omega\mapsto 2^{\mathfrak{c}}$), to
$\bar{g}\colon\beta\omega\to 2^{\mathfrak{c}}$ such that
$\bar{g}(\bigcup_{d_{n}\in D}B_{d_{n}})=D$, i.e.
$\bar{g}(B_{d_{n}})=\textrm{const.}$ for any $d_{n}\in D$. Notice that
$h\colon\beta\omega\setminus\omega\to~{}2^{\mathfrak{c}}$ is onto. We know
that $\beta\omega\setminus\omega=st(P(\omega)/fin)$, (see [11] for details).
By Stone’s Theorem, (see e. g. [6, Theorem 1.1, p. 63]), we have
$st(P(\omega)/fin)$ is isomorphic to $P(\omega)/fin$.

Now let $\varphi_{\alpha}$ be a projection $2^{\mathfrak{c}}$ into
$\alpha$-axe of $2^{\mathfrak{c}}$. Let
$C^{\varepsilon}_{\alpha}=\varphi^{-1}_{\alpha}(\\{\varepsilon\\})$ for any
$\varepsilon\in\\{0,1\\}$ and $\alpha\in 2^{\omega}$.

Obviously $C^{0}_{\alpha}\cap C^{1}_{\alpha}=\emptyset$ and
$C^{0}_{\alpha}\cup C^{1}_{\alpha}=2^{\mathfrak{c}}$ for any $\alpha\in
2^{\omega}$ and $C^{\varepsilon}_{\alpha}\cap
C^{\varepsilon^{\prime}}_{\beta^{\prime}}\not=\emptyset$ for any
$\varepsilon,\varepsilon^{\prime}\in\\{0,1\\}$ and distinct $\alpha,\beta\in
2^{\omega}$. Thus, $\\{(C^{0}_{\alpha},C^{1}_{\alpha})\colon\alpha\in
2^{\omega})\\}$ is an independent family in $2^{\omega}$.

Now, since $h$ is onto, then $2^{\omega}\hookrightarrow P(\omega)/fin$ is an
embedding. Hence $P(\omega)/fin$ contains an independent family of cardinality
continuum. Let $f(C^{\varepsilon}_{\alpha})\\\ =[D^{\varepsilon}_{\alpha}]$.
By Proposition 1, there is an embedding $P(\omega)/fin\hookrightarrow
P(\omega)/\triangle_{d}$. Fix the sequence $\\{k_{m}\colon k\in\omega\\}$ such
that $\lim|[k_{m},k_{m+1})|=\infty$ and

$\lim_{m\to\infty}\frac{|[k_{m},k_{m+1})|}{k_{m+1}}=\frac{1}{2}$

and take mapping $[D^{\varepsilon}_{\alpha}]\mapsto[\bigcup_{m\in
D^{\varepsilon}_{\alpha}}[k_{m},k_{m+1})]$ for any $\varepsilon\in\\{0,1\\}$
and $\alpha\in 2^{\omega}$. Taking $A^{\varepsilon}_{\alpha}=\bigcup_{m\in
D^{\varepsilon}_{\alpha}}[k_{m},k_{m+1})$ we obtain that
$\\{(A^{0}_{\alpha},A^{1}_{\alpha})\colon\alpha\in 2^{\omega}\\}$ is the
required family.

###### Theorem 3

There exists an embedding $\phi\colon H(2^{\mathfrak{c}})/\triangle_{H}\to
P(\omega)/\triangle_{d}$ such that $d(\phi(a))=\mu(a),$ where $a\in
H(2^{\mathfrak{c}})/\triangle_{H}$.

Proof. By Lemma 9, there exists an independent family
$\\{(A^{0}_{\alpha},A^{1}_{\alpha})\colon\alpha\in~{}2^{\omega})\\}$ of
properties (I) and (II).

For any $a\in H(2^{\mathfrak{c}})/\triangle_{H}$ fix exactly one $G_{\delta}$
set $\bar{A}$ such that $a=[\bar{A}]$ and pick a sequence $\\{k_{m}\colon
m\in\omega\\}$ such that $\lim|[k_{m},k_{m+1})|=\infty$ and

$\lim_{n\to\infty}\frac{|[k_{m},k_{m+1})|}{k_{m+1}}=\frac{1}{2}.$

Let $\varphi\colon 2^{\mathfrak{c}}\to 2^{\omega}$ be a continuous map such
that $\mu(\bar{A})=\lambda(\varphi(\bar{A}))$ for any $\bar{A}\in
2^{\mathfrak{c}}$.

Let
$\mathbb{P}_{\bar{A}}=\\{<\bar{A},n,s,D>,n\in\omega,s\in[\omega]^{<\omega},D\in[2^{\omega}]^{<\omega},D\textrm{
is Boolean algebras}\\}$.
Say that $<\bar{A},n,s,D>\leqslant<\bar{A},n^{\prime},s^{\prime},D^{\prime}>$
iff
(i) $n\geqslant n^{\prime},s\supseteq s^{\prime},D\supseteq D^{\prime}$,
(ii) $s\cap(\sup s^{\prime}+1)=s^{\prime}$ and $\sup s=k_{m}$ for some
$m\in\omega$,
(iii) if there is $m^{\prime}<m$ such that

$\forall_{s\in D}\ \forall_{k>k_{m^{\prime}}}\
\forall_{\varepsilon\in\\{0,1\\}}\ \Big{|}\frac{|A^{\varepsilon}_{d}\cap
k|}{k}-\frac{1}{2^{|d|}}\Big{|}<\frac{1}{n}$

and $\bigcap_{d\in D^{\prime}}A^{\varepsilon}_{d}\not=\emptyset$ for any
finite subfamily $D^{\prime}\subset D$, then

$\forall_{s\in D}\ \forall_{k_{m^{\prime}}\leqslant k\leqslant k_{m}}\
\forall_{\varepsilon\in\\{0,1\\}}\ \Big{|}\frac{|s\cap\bigcap_{d\in
D^{\prime}}A^{\varepsilon}_{d}\cap k|}{k}-\lambda(\varphi(\bar{A})\cap
d)\Big{|}<\frac{1}{2^{|d|}n}$

Now consider

$B_{d}(\bar{A})=\\{<\bar{A},n,s,D>\colon d\in D\\}\textrm{ and
}B_{m}(\bar{A})=\\{<\bar{A},n,s,D>\colon n\geqslant m\\}.$

Both sets are dense. Let $G(\bar{A})$ be $\\{B_{d}(\bar{A})\colon d\in
2^{\omega}\\}\cup\\{B_{m}(\bar{A})\colon m\in\omega\\}$-generic. Such
$G(\bar{A})$ exists because our family of subsets is dense in
$\mathbb{P}_{\bar{A}}$ is countable.

Let

$C(\bar{A})=\bigcup\\{s\colon<\bar{A},n,s,D>\in G(\bar{A})\\}.$

and $\\{C(\bar{A})\colon[\bar{A}]=a,a\in
H(2^{\mathfrak{c}})/\triangle_{H}\\}$.
To complete the proof, it is enough to show that
(a) $d(C(\bar{A})\cap C(\bar{A}^{\prime}))=0$ and $d(C(\bar{A})\cup
C(\bar{A}^{\prime}))=1$ for $a^{\prime}=-a\in
H(2^{\mathfrak{c}})/\triangle_{H}$, where $[A^{\prime}]=a^{\prime}$,
(b) $d(\bigcap_{n}^{i=1}C(\bar{A}_{i})\doteq
C(\bigcap_{i=1}^{n}\bar{A}_{i}))=0$, where $\doteq$ means the symmetric
difference.
We will prove (b) only. The proof of (a) is similar.

Fix $\eta>0$. Let $D^{\prime}\subset D$ be a finite family of non-comparable
elements in $2^{\omega}$ such that

$\lambda(\bigcup D^{\prime}\doteq
C(\bar{A}_{1}\cap...\cap\bar{A}_{n}))<\frac{\varepsilon}{3}$

and for any $d\in D^{\prime}$

$|\lambda(C(\bar{A}_{i})\cap d)-\frac{1}{2^{|d|}}|<\frac{\eta}{3}.$

By (iii) we have

$\forall_{1\leqslant i\leqslant n}\ \forall_{\varepsilon\in\\{0,1\\}}\
\exists_{k^{i}_{m}}\ \forall_{k\leqslant k^{i}_{m}}\
\Big{|}\frac{C(\bar{A}_{i})\cap A^{\varepsilon}_{d}\cap
k}{k}-\lambda(\varphi(\bar{A}_{i})\cap d)\Big{|}<\frac{\eta}{3}$

Since

$\forall_{d\in D^{\prime}}\ \forall_{k>\max_{i}k^{i}_{m}}\
\forall_{\varepsilon\\{0,1\\}}\Big{|}\frac{C(\bar{A}_{1})\cap...\cap
C(\bar{A}_{n})\cap\bigcap_{d\in D^{\prime}}A^{\varepsilon}_{d}\cap
k}{k}-\lambda(\varphi(\bar{A})\cap d)\Big{|}<\frac{\eta}{3},$

then for each $k>K$, whenever $K$ is sufficiently large

$\Big{|}\frac{C(\bar{A}_{1})\cap...\cap C(\bar{A}_{n})\cap\bigcap_{d\in
D^{\prime}}A^{\varepsilon}_{d}\cap
k}{k}-\frac{C(\bar{A}_{1}\cap..\cap\bar{A}_{n})\cap A^{\varepsilon}_{d}\cap
k}{k}\Big{|}<\eta.$

The proof is complete.

## References

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Ryszard Frankiewicz, Silesian University of Technology, Gliwice, Poland.
e-mail<EMAIL_ADDRESS>

Joanna Jureczko, Wrocław University of Science and Technology, Poland
e-mail<EMAIL_ADDRESS>


# Detection of Tool based Edited Images from Error Level Analysis and
Convolutional Neural Network

Abhishek Gupta
Engineer
University of Mumbai
Mumbai
<EMAIL_ADDRESS>
Raunak Joshi
Research Scholar
University of Mumbai
Mumbai
<EMAIL_ADDRESS>
Ronald Laban
Assistant Professor
SJCEM
Palghar
<EMAIL_ADDRESS>

###### Abstract

Image Forgery is a problem of image forensics and its detection can be
leveraged using Deep Learning. In this paper we present an approach for
identification of authentic and tampered images done using image editing tools
with Error Level Analysis and Convolutional Neural Network. The process is
performed on CASIA ITDE v2 dataset and trained for 50 and 100 epochs
respectively. The respective accuracies of the training and validation sets
are represented using graphs.

_K_ eywords Convolutional Neural Network $\cdot$ Error Level Analysis $\cdot$
Image Forensics $\cdot$ Image Tampering

##  1 Introduction

The field of Deep Learning[1] has subsequently made groundbreaking research in
various areas of vision, linguistics and audio but the basis of it can be
traced back from the days of Machine Learning[2]. Basically the idea revolves
around the prognostication of patterns from the data with differences. Machine
Learning was certainly derived from the inferential statistic and later grew
to become an independent area of research. The machine learning is derived in
primary divisions of classification and regression, where we focus on
classification being the problem of this paper. Binary Classification[3] is
the precise type of classification that we want to focus on. The binary
classification starts with various algorithms in field of machine learning,
starting from logistic regression[4] to discriminant analysis[5], support
vectors[6], nearest neighbors[7], bagging[8], boosting ensemble[9] and
stacking generalization[10] methods, but the problem we focus on is much more
in depth. The need of deep learning is evident as many state of the art
machine learning algorithms we want to use are going to fall short on
performance for our problem. The area of deep learning works with more samples
of data and in more detailed fashion, almost replicating the human tendencies.
The deep learning works with the concept of artificial neural network that
replicates the biological neuron available in human brain. It is capable of
deriving patterns with great depth and create quite a difference when compared
with the tradition machine learning algorithms. The learning representations
in done with the help of forward propagation[11] where the features are learnt
by the network with the help of weights and arbitrarily declared biases. These
later are given to activation functions for deeper learning and minimizing the
load on the network. Later the loss is computed after completion of one
forward propagation and it is optimized in the backward propagation[12]
process where a loss optimizer is used that computes the gradients of the loss
and retraces to weights for better learning. This entire process is considered
as one epoch of training. This process is repeated several times till the
deeper representation of the patterns from the data is done. The problem we
focus on uses computer vision with deep learning, where a specialized neural
network specially designed for vision problems is used, known as Convolutional
Neural Network[13]. This recognizes patterns from the images considering the
number of channels. The process starts by assigning various filters in the
learning process that help the network to get greater details of the image.
These are termed as the features for the network which are also compressed in
dimensions and increased in depth with a fashion to consider only the
important features. These features are later flattened and treated as an
ordinary neural network to work with the output layer. The problem that we
focus on is classification of an image after using editing tools. The editing
tools in the area of deep learning[14] can be used but this is about detection
of the edited image using deep learning. The similar concepts will be
elaborated in further sections of the paper to work with problem of the forged
images with editing tools.

##  2 Methodology

This section of paper focuses on the workflow design and implementation of the
problem. The problem revolves around the detection of tampered image with
error level analysis given to deep learning model. The dataset is the first
and most important portion of the implementation.

###  2.1 Dataset

The dataset we used for this problem is CASIA dataset. The CASIA ground-
truth[15] dataset is prominent version that contains 8 categories of images.
CASIA ground-truth dataset has an extension that is being used as the tampered
images dataset. Known as CASIA ITDE v.2[16], basically the dataset contains 2
prime classes, authentic images and tampered images. These tampered images are
done using photoshop image editing tool.

###  2.2 Error Level Analysis

Image forensics[17] is the branch where Error Level Analysis[18] also
abbreviated as ELA is used for identification of the image portions with
different compression levels is done. The representation of this can be done
for detection of the tampered images using the editing tool. This can
practically turn out to be a very good preprocessing step before proceeding
with model. The authentic image can be observed below

Figure 1: Authentic Image

The application of the ELA can be done. It also considers a threshold value
that determines the quality of image that highlights the edges. 90% is the
quality value used which can be seen in the figure 2.

Figure 2: Authentic Image under ELA

The tampered images in similar fashion can be observed, the image and its ELA
version.

Figure 3: Tampered Image

The tampered image under ELA can be seen in figure 4.

Figure 4: Tampered Image under ELA

These images certainly point out the details from the dataset and their
respective divisions. The images are used as the data for training the
network.

###  2.3 Network

The network used is a Convolutional Neural Network that has many elements
involved. The input layer, hidden layers and output layer are the primary
subdivisions of the entire structure.

Figure 5: Network used for Implementation

The input images are given to a 2-Dimensional Convolutional Layer which is
responsible for feature extraction from the image by using a set of filters.
The layer uses 32, 5x5 dimensional set of filters. ReLU[19] is the activation
function used which is basically rectified linear unit. Similarly one more
convolutional 2-dimensional layer with 32, 5x5 filters are used along with
ReLU activation. The parameters learnt by first layer are 2432. The parameters
learnt by second layer are 25632. Followed by the 2 convolutional layers,
2-dimensional max pooling[20] layer is used. This specifically removes the
unnecessary feature information and reduces the dimension of the activation
layer. Parameters learnt from this layer are zero as main motive of the layer
is to reduce the dimension. Although the dimensions after reduction keep the
adequate amount of information, very subsequent amount of information can be
discarded because it affects the bias of the network. Much more information
when held in the process of forward propagation can be responsible for poor
learning representations. Even in the later stages the network during the
backward propagation can run into vanishing gradient[21] problem where
calculating the gradients with loss optimizers becomes a daunting task. In
such a scenario Dropout[22] layers can be used. They discard a lot of useless
information making the networks learn better. Certain amount of information to
be dropped from a neuron needs to be specified. This amount is specified in
percentages. The threshold we used is 25%, and parameters learnt are zero.
Even the dimension of the layers stay the same in this layer. Now we flatten
the entire conv layers for further process. Now the process works like an
ordinary neural network. The flattened layer is passed to ReLU layer with 256
hidden neurons. This layer learns 29491456 parameters. Again one dropout layer
is applied with 50% threshold. In the last layer we use a softmax[23]
activation output layer that gives the probabilities of both classes. Since
this is a binary classification problem, we could’ve used sigmoid[24]
activation, but we used softmax as we wanted to distinguish between the
confidence attained in both the classes after prediction. The backward
propagation of the network uses binary cross-entropy[25] as the loss function
and Adam[26] as loss optimizer function. The model has been trained for
different set of epochs to observe the effect in different ways. One time for
50 epochs and other time for 100 epochs. These yield different observations
which we will see in the Results section of this paper.

###  2.4 Results

The result section gives how well the implementation has taken the lead. The
model is defined by the 2 metrics, viz. Accuracy and Loss. Accuracy focuses on
how well the model has learnt from the data with training set and validation
set. Loss focuses on its reduction rate, closer to zero, better the loss. This
is also calculated for training and validation.

Figure 6: Accuracy for 50 Epochs

The accuracy for 50 epochs of training and validation can be seen in figure 6.
The exact numbers for training set is 94.33% and validation set is 92.56%.
Similarly the loss values for training and validation set can be seen in
figure 7.

Figure 7: Loss for 50 Epochs

The exact values for the loss of training and validation for 50 epochs is
16.31% and 19.63%. Similar way of analysis for 100 epochs can be seen for
training and validation accuracy in figure 8.

Figure 8: Accuracy for 100 Epochs

The exact values for the accuracy over 100 epochs with training and validation
sets are 96.13% and 92.20% respectively. This same can be done for loss values
over 100 epochs and can be seen in figure 9.

Figure 9: Loss for 100 Epochs

The exact values for the loss over 100 epochs with training and validation
sets are 10.27% and 20.50% respectively.

##  3 Conclusion

The tampering of images is done using image editing tools and this is a form
of image forensics problem which in this paper is solved by leveraging deep
learning. The Error Level Analysis technique is applied to every single image
in the training data of CASIA that creates a good distinction between
authentic and tampered images using editing tools. Later a simple
Convolutional Neural Network is used to classify the 2 separate classes.
Result section of the paper gives good insights on it and this is a very
simple point we tried working with and later can be used to yield more
effective applications for which we would be more than glad if this paper is
used as a reference material.

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# Persistent Upflows and Downflows at Active Region boundaries Observed by
SUTRI and AIA

Yuchuan Wu  National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100101, China  School of Astronomy and Space Science, University of
Chinese Academy of Sciences, Beijing 100049, China  Zhenyong Hou  School of
Earth and Space Sciences, Peking University, Beijing, 100871, China  Wenxian
Li  National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100101, China  Key Laboratory of Solar Activity and Space Weather, National
Space Science Center, Chinese Academy of Sciences, Beijing 100190, China
Xianyong Bai  National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100101, China  Key Laboratory of Solar Activity and Space
Weather, National Space Science Center, Chinese Academy of Sciences, Beijing
100190, China  School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China  Yongliang Song  National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101,
China  Key Laboratory of Solar Activity and Space Weather, National Space
Science Center, Chinese Academy of Sciences, Beijing 100190, China  Xiao Yang
National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100101, China  Key Laboratory of Solar Activity and Space Weather, National
Space Science Center, Chinese Academy of Sciences, Beijing 100190, China
Ziyao Hu  National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100101, China  School of Astronomy and Space Science, University of
Chinese Academy of Sciences, Beijing 100049, China  Yuanyong Deng  National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101,
China  School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China  Key Laboratory of Solar Activity and Space
Weather, National Space Science Center, Chinese Academy of Sciences, Beijing
100190, China  Kaifan Ji  Yunnan Observatories, Chinese Academy of Sciences,
Kunming, 650011, China Zhenyong Hou; Xianyong Bai<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Upflows and downflows at active region (AR) boundaries have been frequently
observed with spectroscopic observations at extreme ultraviolet (EUV)
passbands. In this paper, we report the coexistence of upflows and downflows
at the AR boundaries with imaging observations from the Solar Upper Transition
Region Imager (SUTRI) and the Atmospheric Imaging Assembly (AIA). With their
observations from 2022 September 21 to 2022 September 30, we find 17
persistent opposite flows occurring along the AR coronal loops. The upflows
are prominent in the AIA 193 Å images with a velocity of 50-200 km s-1, while
the downflows are best seen in the SUTRI 465 Å and AIA 131 Å images with a
slower velocity of tens of kilometers per second (characteristic temperatures
(log $T$(K)) for 193 Å, 465 Å and 131 Å are 6.2, 5.7, 5.6, respectively). We
also analyze the center-to-limb variation of the velocities for both upflows
and downflows. The simultaneous observations of downflows and upflows can be
explained by the chromosphere-corona mass-cycling process, in which the
localized chromospheric plasma is impulsively heated to coronal temperature
forming a upflow and then these upflows experience radiative cooling producing
a downflow with the previously heated plasma returning to the lower
atmosphere. In particular, the persistent downflows seen by SUTRI provide
strong evidence of the cooling process in the mass cycle. For upflows
associated with open loops, part of the plasma is able to escape outward and
into the heliosphere as solar wind.

Sun: chromosphere — Sun: transition region — Sun: corona — Sun: UV radiation

##  1 Introduction

From EUV and X-ray imaging observations, quasi-periodic propagating
disturbances (PDs) along the fan-structures at active region (AR) boundaries
have been observed for decades (Berghmans & Clette, 1999). The upward
propagating disturbances (UPDs) are usually quite common and have been
observed in polar plumes (e.g., Ofman et al., 1997, 1999; DeForest & Gurman,
1998; Tian et al., 2011b; Jiao et al., 2015, 2016) and ARs (e.g., Berghmans &
Clette, 1999; De Moortel et al., 2000; Stenborg et al., 2011; Uritsky et al.,
2013; Huang et al., 2021). The UPDs always has a fast propagation speed of
50-200 km s-1, while downward propagating disturbances (DPDs) are slower with
a speed of tens of km s-1 (McIntosh et al., 2012). A temperature dependence of
the existence of UPDs or DPDs and their velocities is observed at AR
boundaries. For example, Kamio et al. (2011) and McIntosh et al. (2012) found
that the UPDs appear in the hot 193 Å passband with a fast velocity of about
100 km s-1, while the sporadic slow DPDs are formed in the cool passbands such
as 131 Å with a speed of 15 km s-1. McIntosh et al. (2012) also found the
coexistence of upflows and downflows in the 171 Å observations.

Blue and red shifts of spectral lines, which usually coexist with PDs, have
been frequently reported from spectroscopic observations (e.g., De Pontieu &
McIntosh, 2010; Tian et al., 2011a, 2012). The corresponding velocities can be
estimated from the blue and red shifts. Tian et al. (2011a) applied the single
Gaussian fit to the observed coronal line profiles and yielded a blue shift of
10-50 km s-1. However, a more detailed analysis using double-Gaussian fitting
technique (e.g., Peter, 2010; Kitagawa & Yokoyama, 2015) and red-blue (RB)
asymmetry analysis (e.g., De Pontieu et al., 2009; De Pontieu & McIntosh,
2010) suggests that the coronal emission consist of two components, i.e. a
primary component for stationary background and a secondary component
associated with high-speed upflows (Tian et al., 2011a; Brooks & Warren,
2012). The primary component is often blueshifted but with a slower speed of
about 10 km s-1, while the secondary component corresponds to a much faster
velocity of 50–150 km s-1, sometime reaching 200 km s-1 (De Pontieu &
McIntosh, 2010; Tian et al., 2011a). In addition, a clear center-to-limb
variation of doppler shifts is observed (Tian et al., 2012; Ghosh et al.,
2019; Rajhans et al., 2023). The doppler shifts are close to 0 when the AR is
located at the limb of the Sun and reach maximum as the AR rotates to the
center of solar disk (Ghosh et al., 2019; Rajhans et al., 2023). The
dependence of the red and blue shifts on formation temperature has also been
studied. An average red shift is normally observed in spectral lines with
lower formation temperatures and blue shift for lines with higher formation
temperatures (e.g., Peter & Judge, 1999; Tripathi et al., 2009; Dadashi et
al., 2011; Xia et al., 2004; Raju, 2009; Fu et al., 2014). Xia et al. (2004)
found that the transition from red shift to blue shifts occurs at a
temperature (log $T$(K)) of 5.8 for coronal holes and 6.0 for quiet Sun. For
AR fan-like structures, the transition temperature (log $T$(K)) is 6.0 (Warren
et al., 2011).

DPDs and red shifts are explained by downflows (McIntosh et al., 2012), while
there are two explanations for UPDs and blue shifts, upflows (e.g., Sakao et
al., 2007; Harra et al., 2008; Tian et al., 2011a) and slow-mode magneto-
acoustic waves (e.g., Wang et al., 2009; de Moortel, 2009; Nishizuka & Hara,
2011), and now researchers believe that both upflows and waves exist (Tian et
al., 2021). Thus co-spatial UPDs and DPDs (labeled as bidirectionally
propagating disturbances, BPDs, hereafter) can be explained by co-spatial
upflows and downflows, and their temperature-dependence imply a chromosphere-
corona mass cycle process. As suggested by McIntosh et al. (2012), the plasma
in the chromosphere is impulsively heated to transition region and coronal
temperature, exhibiting as high-speed upflows. Then these upflows experience
radiative cooling process and the previously heated materials will slowly
return to the chromosphere, producing slow downflows. A similar scenario of
mass cycling has also been proposed by Marsch et al. (2008) and Young et al.
(2012). Additionally, for UPDs in open loops associated with open-field lines,
part of the plasma can escape outward and into the heliosphere as fast solar
wind. For example, Uritsky et al. (2023) reported self-similar outflows of
hundreds of km s-1 above a polar coronal hole, and Kumar et al. (2023)
reported several jets in AR during a failed eruption. These events are
believed to be a mixture of hot plasma and Alfv$\rm\acute{e}$n waves, and can
be a source of fast solar wind (Raouafi et al., 2023; Uritsky et al., 2023;
Kumar et al., 2022, 2023). Thus the study of the BPDs are highly related to
our understanding of the mass and energy transport mechanism in the solar
atmosphere as well as the formation of the upper atmospheric heating.

By the analyses of observations from Atmospheric Imaging Assembly (AIA, Lemen
et al., 2012) and the Solar Upper Transition Region Imager (SUTRI, Bai et al.,
2023; Wang et al., 2023), we present the spatial and temporal properties of
the upflows and downflows occurring at the AR boundaries and their appearance
in the hot and cool passpands. In particular, SUTRI provides full-disk solar
observations at Ne VII 465 Å line formed at a temperature of 0.5 MK in
transition region, which has been rarely explored. In Section 2 we describe
the observations and analysis methods. In Section 3 we present the
characteristics of the BPD events and discussions. Finally, we summarize our
findings in Section 4.

##  2 Observations and Analysis

In this research, we mainly analyzed the imaging observations taken by SUTRI
on board the Space Advanced Technology demonstration satellite (SATech-01) and
the AIA on board the Solar Dynamics Observatory (SDO, Pesnell et al., 2012).
SUTRI provides narrow-band imaging of full solar disk in the Ne vii 465 Å
passband, which is formed at a rarely sampled temperature of about 0.5 MK (log
$T$(K)$\approx$5.7, Tian, 2017). The normal cadence of SUTRI observations is
30 s and the sampling resolution is $1.22^{\prime\prime}$/pixel. Since about
1/3 of SATech-01’s orbit period ($\sim$96 min) is in the earth eclipse,
SUTRI’s maximum continuous observation time is about an hour. AIA provides
full-disk solar observations in seven different EUV passbands, i.e., 94 Å (Fe
XVIII), 131 Å (Fe VIII for AR, Fe XXI for flare), 171 Å(Fe IX), 193 Å (Fe XII
for AR, Fe XXIV for flare), 211 Å (Fe XIV), 304 Å (He II), and 335 Å (Fe XVI),
spanning a temperature range from about $6\times 10^{4}$ K to $2\times 10^{7}$
K. For our analysis, we mainly used 304 Å, 131 Å, 171 Å, and 193 Å passbands,
with characteristic temperatures (log $T$(K)) of 4.7, 5.6, 5.8, and 6.2,
respectively. The AIA images have a sampling resolution of
$0.6^{\prime\prime}$/pixel and a cadence of 12 s.

PDs in ARs are sometimes covered by the surrounding dynamic structures and are
hard to see in the EUV observations. To better reveal the faint propagating
signatures, Tian et al. (2011b) and McIntosh et al. (2012) used the detrended
intensities for their analyses. Following the method of Tian et al. (2011b)
and McIntosh et al. (2012), we obtained the detrended intensities by
subtracting a $X$-minute running average from the raw intensities. To
highlight PDs with different velocities (McIntosh et al., 2012), we choose a
$X$ of 8 and 20, respectively, to enhance the relative contribution of the
faster and slower PDs to the total emissions.

We analyze SUTRI and AIA observations of AR 13105, 13106, and 13107 from 2022
September 21 to 2022 September 30, during which time the ARs rotate from
eastern limb to western limb. By studying the detrended movies, we can easily
discern the persistent UPDs or/and DPDs occurring along the AR coronal loops
depending on the temperature of the diagnostic passband used. For event with
simultaneous appearance of UPDs and DPDs on the same structure, we define it
as BPD and include it in our studies. Then we perform the space–time analysis
to investigate the evolution of the PDs along the coronal loop for the four
AIA passbands (304, 131, 171, and 193 Å) and SUTRI observations (465 Å).

Table 1: Detailed information of the BPDs.

BPD | Time | Location | Velocity (km s-1) | SUTRI | AIA
---|---|---|---|---|---
ID | (UT) | (arcsec) | DPD | UPD | 465 Å | 304 Å | 131 Å | 171 Å | 193 Å
1 | 2022-9-21 00:19–01:10 | (-796,-264) | 35$\pm$7 | 171$\pm$30 | d |  | d | u | u
2 | 2022-9-21 00:19–01:10 | (-785,-260) | 44$\pm$6 | 143$\pm$18 | d | d | d | d | u
3 | 2022-9-21 17:40–18:32 | (-688,-275) | 65$\pm$5 | 163$\pm$24 | d | d | d,u | d,u | d,u
4 | 2022-9-23 10:42–11:10 | (-427,-304) | 36$\pm$14 | 153$\pm$16 | d |  | d |  | u
5 | 2022-9-24 10:23–11:10 | (-572,-492) | 34$\pm$6 | 180$\pm$40 | d |  | d | d,u | d,u
6 | 2022-9-24 19:54–20:37 | (-116,-405) | 25$\pm$4 | 139$\pm$23 | d |  | d | d,u | u
7 | 2022-9-24 21:26–22:11 | (-88,-393) | 41$\pm$5 | 71$\pm$9 | d |  | d | d | u
8 | 2022-9-24 23:01–23:46 | (-129,-317) | 48$\pm$6 | 141$\pm$25 | d | d | d | d | d,u
9 | 2022-9-25 08:26–09:13 | (25,-259) | 31$\pm$3 | 128$\pm$21 | d |  | d |  | d,u
10 | 2022-9-25 08:26–09:13 | (24,-259) | 29$\pm$3 | 113$\pm$16 | d,u |  | d,u | d,u | u
11 | 2022-9-27 06:13–06:54 | (-57,-452) | 38$\pm$6 | 113$\pm$18 | d |  | d | d | u
12 | 2022-9-27 07:47–08:28 | (393,-394) | 48$\pm$8 | 86$\pm$8 | d | d | d | d | d,u
13 | 2022-9-27 12:32–12:54 | (475,-349) | 31$\pm$9 | 100$\pm$15 | d |  | d |  | u
14 | 2022-9-27 17:16–17:55 | (527,-238) | 28$\pm$5 | 236$\pm$40 | d | d | d |  | u
15 | 2022-9-27 23:34–9-28 00:14 | (527,-238) | 48$\pm$7 | 104$\pm$10 | d |  | d | d | u
16 | 2022-9-28 05:53–06:48 | (612,-270) | 32$\pm$6 | 149$\pm$18 | d | d | d |  | u
17 | 2022-9-29 05:34–06:28 | (725,-248) | 19$\pm$3 | 64$\pm$8 | d | d | d |  | u

Note. — We use the coordinates of the footpoints of the AR loops as the
locations of the BPDs and give them in the third column. In the last five
columns, ‘d’ and ‘u’ represent, respectively, the presence of a DPD and UPD in
each passband.

Figure 1: Solar observations on 2022 September 21 (top row) and September 24
(bottom row). The left column shows the full-disk SUTRI images with a square
region showing the ARs where the BPDs occur. The middle and right columns show
the zoomed-in images of the square regions in SUTRI 465 Å and AIA 193 Å
passbands, respectively. The green rectangles outlined in the middle and right
panels represent the location of the space–time plots studied in Figures 2 and
3.
(Animations of BPD 1 and BPD 8 are available in the online journal, showing
the evolution of these two events. Note that animation of BPD 1 shows
observation data of 2022 September 21 00:19-00:54 UT, and animation of BPD 8
shows observation data from 2022 September 24 23:01-23:46 UT. Animation of BPD
1 has a shorter duration compared with BPD 1 in Table 1, because we cut the
last frames in which part of SUTRI observations are blocked due to the earth
eclipse for a better vision effect. Animation for each event includes four
panels, the first row shows original images of SUTRI 465 Å and AIA 193 Å
passbands, and the second row shows corresponding 20-minute detrended images.
The green rectangles outlined in all panels represent the location of the
space–time plots studied in Figures 2 and 3.)  Figure 2: From top to bottom:
Space–time plots of the detrended intensity signal for BPD 1 in the AIA 304 Å,
SUTRI 465 Å, AIA 131 Å, 171 Å, and 193 Å passbands along the path identified
in Figure 1. The left and right panels show the 8-minute and 20-minute
detrended intensity signals, respectively. The black dashed lines show the
identified UPDs and DPDs. Figure 3: Same as Figure 2, but for BPD 8. Figure 4:
Scatterplots of downflow (left panel) and upflow (right panel) velocities as a
function of radius vector. The error bars represent the uncertainties of the
velocities.

##  3 Results and Discussion

We identify 17 BPD events on the coronal structures and the detailed
information, i.e., occuring time, locations, propagating velocities, and
appearance in SUTRI and AIA images, are listed in Table 1. The ARs appeared on
the east limb on 2022 September 21 and moved to the disk center following the
rotation of the Sun on 2022 September 24 and disappeared behind the western
limb on September 30. By observing the loops structure in AIA 193 Å, we can
see that most of the magnetic geometry of the studied loops are open, and
there are also several events with closed loops. Figure 1 shows the snapshot
of SUTRI and SDO/AIA 193 Å observations for BPD 1 (upper panels) and 8 (lower
panels). BPD 1 is an event within an open loop, while BPD 8 is an event within
a closed loop. From the online supporting movies, we can see clearly the PDs
in the fan-like structure at the boundary of the ARs, i.e., slow down-flowing
plasma streams in SUTRI 465 Å and fast upward motions in the AIA 193 Å
passband. Note that SUTRI’s maximum continuous observation time is about 60
minutes due to the earth eclipse.

Figure 2 and Figure 3 show the space-time plots for BPD 1 and BPD 8,
respectively. The starting point of $Y$-axis in the space-time plots
corresponds to the footpoint of the AR loops. The left panels show the signal
following the removal of an 8-minute running average from the raw intensity
while the right panels show the 20-minute detrended signals. From the figure
we can see the evolution of the PDs along the coronal loop for the AIA 304 Å,
SUTRI 465 Å, 131 Å, 171 Å, and 193 Å passbands from top to bottom. Note that a
running average of two frames was applied to AIA images to improve the signal-
to-noise ratio, especially for 304 and 131 Å. In Figures 2 and 3, the DPDs and
UPDs appear as inclined bright stripes with opposite directions. The left and
right panels show clear difference in the patterning visible. For example, the
inclined black dashed lines indicate an apparent downward motion of material
along the structure in Figure 2 (b2) and (c2), while no obvious signatures are
shown in Figure 2 (b1) and (c1). For upflows in AIA 193 Å, however, the space-
time plots of the 8-minute detrended intensity reveal more signatures than
that of the 20-minute detrended intensity. McIntosh et al. (2012) have
observed the same phenomenon and discussed the explanation. As what is shown
in Figure 2 and 3, upflows are associated with shorter lifetime and downflows
with longer lifetime. For a smaller detrending time which is shorter than
upflows’ lifetime, a single upflow can be well sampled from beginning to its
end. However, only a small portion of a single downflow is sampled during the
lifetime of a single upflow. That is because the intensity variation of
downflows is removed as the background in the detrended intensity plots.
Consequently, for some downflows, they are not so clear or even cannot be
observed in the 8-minute detrended plots. While for a longer detrending time
which is comparable to upflows lifetime, only a part of the upflows can be
sampled from their beginning to the end. So in the 20-minute detrended plots,
fewer upflows can be observed. However, for downflows, they can be sampled
from beginning to the end, so they can be clearly viewed in 20-minute
detrended plots.

The appearance of the BPDs in each passpand is given in the last five columns
of Table 1, from which we found that the downflows are prominent in SUTRI 465
Å and AIA 131 Å while the upflows are best seen in AIA 193 Å. In some events,
we can see the appearance of both downflows and upflows in the same passband.
The above analysis indicates that the flows are temperature-dependent, with
upflow of PDs visible in the hotter passbands while downflows in cooler
passbands; this is consistent with previous results by McIntosh et al. (2012).

Based on the appearances of the downflows in the EUV passbands, we divided the
BPDs into two categories. In category 1, downflows only appear in the low-
temperature passbands such as the SUTRI 465 Å and AIA 131 Å (e.g., BPD 1). On
the other hand, the BPDs in category 2 show downflow signatures both in the
low- and high-temperature passbands (e.g., BPD 8 appearing in AIA 193 Å).
Category 1 includes BPD 1, BPD 4, BPD 13, BPD 14, BPD 16, and BPD 17, and
category 2 includes BPD 2, BPD 3, BPD 5, BPD 6, BPD 7, BPD 8, BPD 9, BPD 10,
BPD 11, BPD 12, and BPD 15.

The projected velocities of the BPDs on the plane-of-sky (POS) can be
estimated by calculating the slopes of the strips, which are indicated by the
dashed line in the space-time maps. When the DPDs/UPDs appeared in multiple
passbands, we choose the passband in which DPDs/UPDs can be most clearly
observed to estimate POS; they are usually SUTRI 465 Å for DPDs and AIA 193 Å
for UPDs. We first define a threshold and select the signals with higher (or
lower, in case of a dark stripe) detrended intensity than the threshold in a
region where a bright stripe can be observed. We choose a suitable threshold
to highlight the stripe and separate it from the background. Then, we use a
linear fitting to get the slope as the estimated velocity. The error of
velocity mainly comes from the estimate of distance, and is estimated by
$\frac{E(\Delta D)}{\Delta T}$, where $\Delta T$ is estimated by the lasting
time of PD, and $E(\Delta D)$ is estimate by $\sqrt{2}RMSE(D)$, where
$RMSE(D)$ means the root mean square error of the distance in the linear
fitting. Then if there are multiple DPDs/UPDs in one event, we calculate their
average velocity. As an example, the POS velocities of DPDs in BPD 1 and BPD
8, as shown in Figure 2 and Figure 3, are usually within the range of 33–57 km
s-1, while the UPDs are always associated with a fast speed, i.e. 121–219 km
s-1. The averaged POS velocities of DPDs and UPDs in BPD 1 (BPD 8) are
35$\pm$7 (40$\pm$6) km s-1and 171$\pm$30 (141$\pm$25) km s-1, respectively.
The estimated POS velocities together with their estimated errors for all the
17 events are shown in Table 1. We can see that the downflow usually has a POS
velocity of 19–65 km s-1, while the upflow velocity ranges from 64 km s-1 to
236 km s-1.

Previous studies have demonstrated that the prevalent UPDs appearing in the
fan-like structures at AR boundaries are related to the high-speed secondary
components (blueward asymmetries) of coronal lines (Tian et al., 2021). In
this work, the velocity of the upward propagating features is estimated to be
64–236 km s-1, which is consistent with the previous results. The velocities
of the downward propagating features are similar to those obtained from the
redshifts of the transition region and coronal lines in the previous studies
(e.g., Tian et al., 2011a, b). From the above analysis it indicates that the
downward and upward propagating features studied in this work are likely
plasma flows propagating with opposite directions along the coronal loop legs
at the AR boundaries.

The above analysis suggests that we are observing the same process with that
from McIntosh et al. (2012); Tian et al. (2021), i.e. the mass-cycling between
the chromosphere and corona. As indicated in Figure 5 of McIntosh et al.
(2012), the heated material associated with highly dynamic spicules (McIntosh
& De Pontieu, 2009; De Pontieu et al., 2011; Tian et al., 2014; Samanta et
al., 2019) move upward from the chromosphere to the corona with a fast
velocity along the coronal loops. The upflow will lead to a high-speed
secondary emission component in coronal lines (Tian et al., 2011a). The
persistent downflows seen in cooler emission are a result of the previously
heated plasma transitioning into the radiative cooling domain and then
returning to the lower atmosphere with a slower speed. Also, we noticed that
for some events in open loops such as BPD 1, upflows share similar
characteristics with self-similar outflows that have been reported by Uritsky
et al. (2023). For these events, the mass-cycling between the chromosphere and
corona may not be a fully closed cycle and a part of plasma in UPDs can escape
outward and into the heliosphere. This process may be a source of fast solar
wind (Raouafi et al., 2023; Uritsky et al., 2023; Kumar et al., 2022, 2023).

We noticed that BPDs events can be divided into two categories: downflows only
appear in the low-temperature passbands in category 1, and appear both in the
low- and high-temperature passbands in category 2. This indicates that some
downflows only contain cold plasma, and other downflows are mixtures of cold
plasma and hot plasma. The ubiquitous downflows seen at a temperature of 0.5
MK in transition region observed by SUTRI provide additional evidence for the
radiative cooling process in the complex chromosphere-corona mass-cycle.

In addition, we investigated the relationship between velocities and the
radius vector (RV, the fractional distance to the limb from the disk center),
to study the center-to-limb variation (CLV) of POS velocities. RV is defined
as, based on Klimchuk (1987); Rajhans et al. (2023), $\rm
RV=\pm\sqrt{x^{2}+y^{2}}/R_{sun}$, where $(x,y)$ is the coordinate of the
footpoint of the coronal loops in units of arc-second and $R_{sun}$ is the
solar radius with a value of 959′′. A value of zero corresponds to disk center
whereas $+1(-1)$ represents the eastern (western) limb. Figure 4 plots the POS
velocities as a function of RV for downflows and upflows. Although the
velocities are in wide scatter, we see a weak CLV variation, i.e., larger POS
velocities are observed as the RV approaches the limb. This can be explained
by taking into account the LOS projection effect. Considering the fact that
the coronal loops hosting the BPDs are located at the AR boundaries, these
loops are likely perpendicular to the line-of-sight (LOS) when the ARs are
close to the solar limb. On the contrary, when the ARs are approaching the
solar disk center, the coronal loops are roughly aligned with the LOS and the
POS velocities of the BPDs are small. Such a result and interpretation is
consistent with the finding of Tian et al. (2012), in which the authors
investigated the velocities of the plasma flows from the CLV of the
spectroscopic profile asymmetries.

##  4 Summary

We report 17 BPDs, consisting of coexisting upflows and downflows, from
imaging observations taken by SATech-01/SUTRI and SDO/AIA. These plasma flows
persistently appear at the AR boundaries as the ARs rotate from the east limb
to the disk center and then to the west limb during the time period of 2022
September 21 to 2022 September 30.

The upflows are prominent in hotter passband, i.e. AIA 193 Å, while the
downflows are best seen in SUTRI 465 Å and AIA 131 Å. In some cases, the
downflows can also be seen in AIA 304 Å, 171 Å, and 193 Å. The velocities of
downflows are estimated to be only tens of kilometers per second, while the
upflows propagate more faster with a velocity of 50-200 km s-1. The
coexistence of downflows and upflows can be explained by a chromosphere–corona
mass-cycling process, in which the local chromospheric plasma is impulsively
heated to the coronal temperature forming an upflow, and then experience a
radiative cooling process producing a downflow with the previously heated
plasma returning to the lower atmosphere. For upflows in open loops, some
plasma can be ejected into space and this makes upflows a source of solar
wind. We also investigate the CLV of the POS velocities for both upflows and
downflows, which is likely due to the LOS projection effect.

##  5 acknowledgement

This work is supported by the National Key R&D Program of China
(2021YFA0718601), the Youth Innovation Promotion Association CAS (2023061) and
the National Natural Science Foundation of China (NSFC, Grant Nos. 12373115).
Z. Y. Hou was supported by NSFC grant 12303057 and China Postdoctoral Science
Foundation No. 2021M700246. AIA is an instrument onboard the Solar Dynamics
Observatory, a mission for NASA’s Living With a Star program. SUTRI is a
collaborative project conducted by the National Astronomical Observatories of
CAS, Peking University, Tongji University, Xi’an Institute of Optics and
Precision Mechanics of CAS and the Innovation Academy for Microsatellites of
CAS.

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