ISSN2188－1200

International Project Research-Workshop (I)WORKSHOPonMathematics for Industry
Basis of Mathematics in nanomedicine
structures and life sensing

Editors: /SAMU3AEKI 7OJCIECH$OMITRZ 3TANISÓAW*ANECZKO-ARCIN:UBILEWICZ -ICHAÓ:WIERZYŷski

九州大学マス・フォア・インダストリ研究所

MI Lecture Note Vol.95 : Kyushu University

International Project Research-Workshop (I)WORKSHOP on Mathematics for Industry

Basis of Mathematics in nanomedicine structures and life sensing

Editors

Osamu Saeki, Wojciech Domitrz, Stanisław Janeczko,
Marcin Zubilewicz, Michał Zwierzyński

A

About MI Lecture Note Series
The Math-for-Industry (MI) Lecture Note Series is the successor to the COE Lecture
Notes, which were published for the 21st COE Program “Development of Dynamic
Mathematics with High Functionality,” sponsored by Japan’s Ministry of Education,
Culture, Sports, Science and Technology (MEXT) from 2003 to 2007. The MI Lecture Note Series has published the notes of lectures organized under the following two
programs: “Training Program for Ph.D. and New Master’s Degree in Mathematics as
Required by Industry,” adopted as a Support Program for Improving Graduate School
Education by MEXT from 2007 to 2009; and “Education-and-Research Hub for
Mathematics-for-Industry,” adopted as a Global COE Program by MEXT from 2008 to2012.
In accordance with the establishment of the Institute of Mathematics for Industry (IMI)in April 2011 and the authorization of IMI’s Joint Research Center for Advanced andFundamental Mathematics-for-Industry as a MEXT Joint Usage / Research Center in
April 2013, hereafter the MI Lecture Notes Series will publish lecture notes and proceedings by worldwide researchers of MI to contribute to the development of MI.
October 2022
Kenji Kajiwara
Director, Institute of Mathematics for Industry

WORKSHOP on Mathematics for Industry

Basis of Mathematics in nanomedicine structures and life sensing
MI Lecture Note Vol.95, Institute of Mathematics for Industry, Kyushu University
ISSN 2188-1200
Date of issue: March 14, 2024
Editors: O samu Saeki, Wojciech Domitrz, Stanisław Janeczko, Marcin Zubilewicz, Michał Zwierzyński
Publisher:
Institute of Mathematics for Industry, Kyushu University
Graduate School of Mathematics, Kyushu University
Motooka 744, Nishi-ku, Fukuoka, 819-0395, JAPANTel +81-(0)92-802-4402, Fax +81-(0)92-802-4405
URL https://www.imi.kyushu-u.ac.jp/

BPreface
The "WORKSHOP on Mathematics for Industry 2023 – Basis of Mathematics in nanomedicine
structures and life sensing" convened during September 25–29, 2023, at Warsaw University ofTechnology, Poland, under the joint auspices of the Faculty of Mathematics and Information
Science, Warsaw University of Technology; Center for Advanced Studies, Warsaw University ofTechnology; and Institute of Mathematics for Industry, Kyushu University, with the support of
the Excellence Initiative: Research University Programme at the Warsaw University of Technology. With the participation of approximately 70 attendees, including researchers, students, andPhD candidates, the workshop served as a nexus for interdisciplinary dialogue and collaboration
between the realms of mathematics and applied sciences.
The workshop program encompassed 25 individual talks and 5 mini-courses, each comprising 3 lectures, spanning a spectrum of topics such as topological data analysis, medical imaging
methods, human genome models, big data, machine learning, cryptography, information geometry, convex optimization, physical models of elastic/plastic bodies and fluids and material
engineering. Delivered by experts from Polish and Japanese institutions, the presentations illuminated the symbiotic relationship between abstract mathematical constructs and real-world
engineering challenges, thereby fostering innovation and knowledge exchange. The accompanying booklet contains comprehensive materials from the workshop prepared by the speakers,
including detailed summaries, presentation slides and references, providing a valuable resource
for continued study of the concepts presented during the event, with hope that it will not only
facilitate the exploration of novel research directions, but also catalyze the establishment ofinternational collaborations between academic environments in Poland and Japan with the goalof leveraging mathematical methodologies to address pressing industrial concerns and societal
needs.
This work was supported by Institute of Mathematics for Industry, Joint Usage/ResearchCenter in Kyushu University (FY2023 Workshop(I) "WORKSHOP on Mathematics for Industry
2023 – Basis of Mathematics in nanomedicine structures and life sensing" (2023b004)).
February 2024i
1

WORKSHOP
on Mathematics for Industry
Basis of Mathematics in nanomedicine
structures and life sensing

25-29 September 2023
Warsaw - Poland
Scientific Committee:

Tomasz Cieślak (Warsaw)
Wojciech Domitrz (Warsaw)
Leon Gradoń (Warsaw)
Naoki Hamada (KLab Inc.)
Yuichi Ike (Kyushu)
Stanisław Janeczko (Warsaw)
Shizuo Kaji (Kyushu)
Kenji Kajiwara (Kyushu)
Miyuki Koiso (Kyushu)
Shigeki Matsutani (Kanazawa)Jan Mielniczuk (Warsaw)
Takashi Nishimura (Yokohama)
Dariusz Plewczyński (Warsaw)
Maria Aparecida Soares Ruas (São Carlos)
Osamu Saeki (Kyushu)
Hiroshi Teramoto (Kansai)
Organizers:

Wojciech Domitrz (Warsaw)
Stanisław Janeczko (Warsaw)
Osamu Saeki (Kyushu)
Marcin Zubilewicz (Warsaw)
Michał Zwierzyński (Warsaw)
Organizing Institutions:

Institute of Mathematics for Industry, Kyushu University
Center for Advanced Studies, Warsaw University of Technology
Faculty of Mathematics and Information Sciences, Warsaw University of Technology

ht tps:/ /wmi2023.mini.pw.edu . p l
Faculty of Mathematics
and Information Sciences
WA R S AW U N I V E R S I T Y O F T E C H N O L O G Y

ii

iii

iv

Lecture:Dinner @ MaIS Faculty

Lecture:Miyuki Koiso

Lecture:Stanisław Janeczko

17:30 – 18:00

18:00 – 22:00

Lecture:Toshizumi Fukui

Lecture:Lucía Ivonne Hernández Martínez

16:45 – 17:15

Coffee break

Coffee break

Lecture:Karol Ćwieka

Lecture:Mariusz Niewęgłowski

15:45 – 16:15

16:15 – 16:45

Lecture:Leon Gradoń

Lecture:Przemysław Biecek

15:00 – 15:30

Lunch

Lecture:Tomasz Cieślak

Lecture:Naoki Hamada

Lunch

Mini course:
Paweł Dłotko 2

Mini course:
Paweł Dłotko 1

Coffee break

13:00 – 15:00

12:30 – 13:00

11:30 – 12:15

Coffee break

Mini course:
Dariusz Plewczyński 1

Mini course:
Shunsuke Ichiki 1

10:15 – 11:00

11:00 – 11:30

Mini course:
Jan Mielniczuk 2

8:30 – 9:00 Paweł Józiak

Mini course:
Jan Mielniczuk 1

Opening of the Workshop

9:00 – 9:10

Tuesday (26.09)

9:15 – 10:00

Registration (up to 11:00)

8:00 – 9:00

Monday (25.09)

Programme

WORKSHOP on Mathematics for Industry 2023

Lunch

Lecture:Przemysław Grzegorzewski

Lecture:Yuichi Ike

Coffee break

Mini course:
Paweł Dłotko 3

Mini course:
Dariusz Plewczyński 2

Mini course:
8:15 – 9:00 Shunsuke Ichiki 2

Wednesday (27.09)

Dinner @ MaIS Faculty

Lecture:Takashi Nishimura

Lecture:Shizuo Kaji

Coffee break

Lecture:Konrad Kisiel

Lecture:Zbigniew Peradzyński

Lunch

Lecture:Shigeki Matsutani

Mini course:
Arimura Hidetaka 2

Coffee break

Mini course:
Arimura Hidetaka 1

Mini course:
Jan Mielniczuk 3

8:30 – 9:00 Piotr Borowik

Lecture:Thursday (28.09)

Friday (29.09)

Lecture:Konstanty Junosza-Szaniawski

Lecture:Bartosz Kołodziejek

Coffee break

Lecture:Naomichi Nakajima

Lecture:Osamu Saeki

Lunch

Lecture:Hiroshi Teramoto

Mini course:
Arimura Hidetaka 3

Coffee break

Mini course:
Dariusz Plewczyński 3

Mini course:
Shunsuke Ichiki 3

Lecture:Kenji Kajiwara■ Mini courses ■ Registration

Contents
Preface ������������������������������������ i
Poster of the Workshop�����������������������������ii
Photos ����������������������������������� iii
Workshop Program������������������������������ iv
Abstracts & Slides for Mini-courses
Paweł Dłotko (Polish Academy of Sciences) �������������������
“Introduction to Topological Data Analysis”

1

Arimura Hidetaka (Kyushu University)��������������������� 63“Medical imaging signatures with topology for cancer”

Shunsuke Ichiki (Tokyo Institute of Technology)�����������������103“Singularity theory and its applications to strongly convex multiobjective optimization
problems”

Abstracts & Slides for Talks
Przemysław Biecek (Warsaw University of Technology) �������������109“Explanatory Model Analysis”

Tomasz Cieślak (Polish Academy of Sciences)������������������125“Linear instability of Prandtl spirals”

Toshizumi Fukui (Saitama University) ���������������������131
“Pseudospheres from singularity theory view-point with a classification of 2-soliton surfaces”

Leon Gradoń (Warsaw University of Technology) ����������������139“Formation of nanostructured functional particles with the spray-drying method”

Przemysław Grzegorzewski (Warsaw University of Technology) ���������153“On comparing distributions with imprecise data”

Naoki Hamada (KLab Inc.)���������������������������167“Brief Introduction to Topology for Multi-objective Optimization”

Yuichi Ike (Kyushu University) ������������������������183“Persistent Homology and Machine Learning”

Stanisław Janeczko (Warsaw University of Technology) �������������193“Exotic shapes of nano-spherical structures - new DNA coding”

Paweł Józiak (Warsaw University of Technology) ����������������213“How to measure data diversity and why is it important?”v

Konstanty Junosza-Szaniawski (Warsaw University of Technology) ��������229“Cryptographic protocol verification - results of EPW project”

Shizuo Kaji (Kyushu University)������������������������239“Synergies of medicine, physics, and mathematics in medical imaging”

Konrad Kisiel (Warsaw University of Technology)����������������247“Plasticity - Modeling and mathematical analysis”

Miyuki Koiso (Kyushu University) �����������������������253“Developable surfaces with curved folds and applications”

Bartosz Kołodziejek (Warsaw University of Technology) �������������261“Learning Permutation Symmetry of a Gaussian Vector”

Shigeki Matsutani (Kanazawa University)��������������������271“Supercoiled structure of DNA and hyperelliptic functions”

Naomichi Nakajima (Waseda University) ��������������������299“Information geometry of positive measures”

Mariusz Niewęgłowski (Warsaw University of Technology)������������309“Multivariate Hawkes processes”

Takashi Nishimura (Yokohama National University) ���������������323“On envelopes created by circle families in the plane (a joint work with Yongqiao Wang)”
Zbigniew Peradzyński (Military University of Technology) ������������333“Calcium waves sustained by calcium influx through mechanically activated channels
in the cell membrane”

Osamu Saeki (Kyushu University) �����������������������355“Generalization of Reeb spaces and application to data visualization”

viWORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Introduction to Topological Data Analysis
Pawel Dlotko
Dioscuri Centre in Topological Data Analysis, IMPAN, Poland
In this mini-course we will explore both theoretical and practical foundations of Topological Data Analysis (TDA) — a field with a number of applications in physical,
natural and social sciences in the intersection between algebraic topology, computational geometry and computational methods. We will cover the basic tools of TDA
including discretization of spaces (in the form of various point cloud-based simplicial,
cubical and general CW-complexes), algorithms to compute homology and persistenthomology and applications of those. We will also explore TDA tools of visualization,
like mapper and ball mapper algorithms. Moreover we will present new tools of Euler Characteristic curves and profiles and show how they can be applied to standard
statistics. All the concepts will be illustrated with real examples. You will also be
required to perform computations on a number of toy and real-world datasets.
References[1] Edelsbrunner, Harrer (2011), Computational Topology: An Introduction[2] P. Dlotko, Computational and applied topology, tutorial, https://arxiv.org/abs/1807.08607

1

Introduction to Topological
Data Analysis
Paweł Dłotko, Dioscuri Centre in TDA, IMPAN,
WORKSHOP on Mathematics for Industry 2023
Politechnika Warszawska, MINI, 25-27 September 2023

Topological Data Analysis▶ Persistent homology,▶ Conventional mapper,▶ Ball mapper,▶ Discrete Morse theory (if time permits),▶ TopoTests (alternative option),▶ On a very intuitive level,▶ with a number of practical examples.

The credo

Data have shape,
shape has meaning,
meaning brings value.

2

We all know this story

Trap of models

It is not possible to adjust an algebraic model to any possible
shape of the data – over-fitting.

Topology and statistics, together▶ Statistics provide a vast collection of tools to summarize
properties of point clouds.
▶ However, there are numerous examples (line Anscombe’s
quartet and Datasaurus dataset presented below) of point
clouds with the same descriptive statistics, but very different
shape.
▶ This is why, in statistics, we should always visualize theconsidered dataset.
▶ It is however not possible to visualize high dimensional data.
▶ That is where the tools from topology came into rescue –topological tools we discuss in this tutorial allow us toestimate if two datasets have similar shape.

3

Anscombe’s Quartet

Anscombe’s Quartet; Same statistics, different shapes
Anscombe, ”Graphs in Statistical Analysis”, American Statistician, 1973.

Datasaurus Dataset

Datazaurus Dozen, Alberto Cairo,
http://www.thefunctionalart.com/2016/08/
download-datasaurus-never-trust-summary.html

TDA pipeline

Topologicaldescriptor
Point cloud4

InferenceSimplicial complexes▶ K is an abstract simplicial complex iff for every A ∈ K and
B ⊂ A, B ∈ K.
▶ Each abstract simplicial complex has its geometricalrealization built from simplices.
▶ In this case, simplices consist of points in a general position.

d
c
b

a

c
a
Dim 0

a
Dim 1

b

a

Dim 2

b
Dim 3

Sample simplicial complexes

Source: Wikipedia, typical use FEM-like methods.

Let the data tell you the story

Topological data analysis:▶ Persistent homology – point-cloud based homology.
▶ Accurate network models to examine landscapes of data,! Stable.
!! No black boxes.
!!! We do not enforce any models of data.

5

What do you see?

What do you see?

▶ We may say that we see a circle,▶ But we really see is 19 points...
▶ ...that may be sampled from a probability distribution
supported at a circle.
▶ Persistent homology is a tool to make this observation precise.
▶ To do so, we need to construct a filtered complex of the point
cloud.
▶ The filtered complex is a nested sequence of subcomplexes - a
way of building a model by adding a sequence of simplices ina number of steps.

What do you see?

6

What do you see?

What do you see?

What do you see?

7Simplicial complexes built from point clouds▶ P = {p1 , . . . , pn }, a finite point cloud with a metric d.

▶ We need a finite, combinatorial representation of the union ofballs.
▶ Rips complex at level ϵ consists of simplices supported in
p0 , . . . , pn if B(pi , 2ϵ ) ∩ B(pj , 2ϵ ) ̸= ∅ for every i, j ∈ {0, . . . , n}.
▶ Čech complex at level ϵ consists of simplices supported inp0 , . . . , pn iff ni=0 B(pi , 2ϵ ) ̸= ∅.

Filtration of Rips complex

4 vertices

Filtration of Rips complex

4 vertices, 1 edge8Filtration of Rips complex

4 vertices, 3 edges, 1 triangle

Filtration of Rips complex

4 vertices, 4 edges, 1 triangle

Filtration of Rips complex

4 vertices, 5 edges, 2 triangles

9Filtration of Rips complex

4 vertices, 6 edges, 4 triangles, 1 tetrahedra

Rips vs Čech

Rips vs Čech

A hole10

Rips vs Čech

r/2

r/2

r/2

r/2
r/2

r/2

In this case Rips complex is a triangle with a boundary, the Čechcomplex is the boundary of a triangleČech complex is topologically accurate▶

ϵp∈P B(p, 2 ) is topologically equivalent to the Čech complex
based on those balls.

▶ Meaning, there exist a continuous deformation from one intoanother.
▶ No tearing, no gluing.

Rips and Čech complexes can grow large

If all points get connected by edges in the complex, we witness
so-called combinatorial explosion. You will encounter it when using
Rips complexes.

11Rips and Čech complexes can grow large

r

 
 
 
For N points, N1 vertices, N2 edges, N3 triangles, ...
This is why we always limit the level (ϵ) and the maximal
dimension of simplices in the complex.

Alpha complexes

Intersecting B(x, r ), for x ∈ X with Voronoi cells of X allows tobuild much smaller complexes that preserve homotopy type ofx∈X B(x, r ).

Be careful with distances (in high dimensions)
1. Concentration of measure (1 − 2ϵ)n ,2. Points in dimension d close to be of the same distance d3 fromeach other in l 1 distance,
3. Manifold hypothesis.

12

From complexes to parameter dependent homology

Simplicial
complex

ChainsCyclesBoundaries

Homology groups

Homology

One connected component, one hole in dimension 1.

Practical exercise 1▶ Please go tohttps://github.com/dioscuri-tda/tutorials,▶ Open PH intro to homology and play with triangulation of atorus.
▶ What are the homology groups of this triangulation?

13

Triangulation of a torus
0

3

4

0

1

8

7

1

2

5

6

2

0

3

4

0

Triangulation of a torus

Persistent homology, under the hood

▶ Let us order simplices according to the minimal ϵ for which
they appear (filtration).
▶ Algorithm to compute (persistent) homology is a version ofGaussian elimination.
▶ If we run it for a prefix of filtration, we will get homology of
the complex composed by simplices in that prefix (asubcomplex of the final complex).
▶ Analyzing the structure of zero and non-zero columns in the
reduced matrix allows us to find generators that are created
and which become trivial as we move along the filtration.

14

Persistence matrix algorithm

4 d
8
9

c

3

7

6

10

b

2
5

a

1

Persistence matrix algorithm

4 dab ac bc cd bd abc
a 1 1
1
1
b 1

8
9

c

3

c

7

6

10

1

1

b

2
5

1
1

d
ab

1
1

ac

1

bc

1

a

1

Persistence matrix algorithm

4 dab ac bc cd bd abc
a 1 1
1
1
b 1

8
9

c

3

c

7

6

10

d
ab

b

2
5

a

1

15

1

1

1
1

1
1

ac

1

bc

1

Persistence matrix algorithm

4 dab ac bc cd bd abc
a 1 1
1
1
b 1

8
9

c

3

c

7

6

10

1

1

b

2
5

1
1

d
ab

1
1

ac

1

bc

1

a

1

Persistence matrix algorithm

4 dab ac bc cd bd abc
a 1 1
1
1
b 1

8
9

c

3

c

7

6

10

1

b

2
5

1

1
1

d
ab

1
1

ac

1

bc

1

a

1

Persistence matrix algorithmbc+
ab ac ac cd bd abc
a 1 1 1
1
1
b 1

4 d
8
9

c

3

c

7

6

10

d
ab

b

2
5

a

1

16

1

1
1

1
1

ac

1

bc

1

Persistence matrix algorithmbc+ac+
ab ac ab cd bd abc
a 1 1
1
b 1

4 d
8
9

c

3

c

7

6

10

1

b

2
5

1
1

d
ab

1
1

ac

1

bc

1

a

1

Persistence matrix algorithmbc+ac+
ab ac ab cd bd abc
a 1 1
1
b 1

4 d
8
9

c

3

c

7

6

10

1

b

2
5

1
1

d
ab

1
1

ac

1

bc

1

a

1

Persistence matrix algorithmbc+ac+
ab ac ab cd bd abc
a 1 1
1
b 1

4 d
8
9

c

3

c

7
10

6

d
ab

b

2
5
a

1

17

1

1
1

1
1

ac

1

bc

1

Persistence matrix algorithm

bc+
ac+
bd+
ab ac ab cd cd abc
a 1 1
1
b 1

4 d
8
9

c

3

c

7

6

10

1

b

2
5

1

1

1

d
ab

1

ac

1

bc

1

a

1

Persistence matrix algorithm

4 d
8
9

c

3

c

7

6

bc+
bd+
ac+
cd+
ab ac ab cd ac abc
1
a 1 1
1
b 1

10

b

2
5

1

1
1

d
ab

1

ac

1

bc

1

a

1

Persistence matrix algorithmbc+ac+
ab ac ab cd
a 1 1
b 1

4 d
8
9

c

3

c

7

6

10

d
ab

b

2
5

a

1

18

1

bd+
cd+
ac+
ab abc

1
1
1

ac

1

bc

1

Interpretation of reduced matrix

1. The reduced matrix gives the persistence intervals.
2. If the column is zero, then it creates a new homology class.
3. If the column is nonzero, then it kills a homology class.

Persistence matrix algorithm

b

bc+
ac+
d ab ac ab cd
1 1
1

c

1

a
4

d

8
3

b

c

a

bd+
cd+
ac+
ab abc

9

c
7
10

6

b
2
5

1a

1

ac

1

bc

1

Persistence matrix algorithm

a
4

d

c
b
2

1a

b

a
b

3

c

1
1

d
ab

d
ab
ac
bc

19

c

d

Persistence matrix algorithm

a
4

d

b

c

d ab
1
1

b

c

b

d ab ac
1 1
1

c

1

a
b

3

c

c
b
2
5

d
ab
ac

1a

bc

Persistence matrix algorithm

a
4

d

a

3

c
b

6
2
5

d
ab
ac

1a

bc

Persistence matrix algorithm

b

bc+
ac+
d ab ac ab
1 1
1

c

1

a
4

d

3

c
7

b

6
2
5
1a

b

a

d
ab
ac
bc

20

c

Persistence matrix algorithm

b

bc+
ac+
d ab ac ab cd
1 1
1

c

1

a
4

d

8
3

c
7

b

6
2
5

b

c

a

1
1

d
ab
ac

1a

bc

Persistence matrix algorithm

b

bc+
ac+
d ab ac ab cd
1 1
1

c

1

a
4

d

8
3

b

c

a

bd+
cd+
ac+
ab

9

c
7

b

6
2
5

1
1

d
ab
ac

1a

bc

Persistence matrix algorithm

b

bc+
ac+
d ab ac ab cd
1 1
1

c

1

a
4

d

8
3
9

c
6

7
10
2
5

1a

b

b

a

c

bd+
cd+
ac+
ab abc

1
1

d
ab

1

ac

1

bc

1

Dim 0: [2,5], [3,6], [4,8]Dim 1: [7.10]

21

Persistence matrix algorithm

b

bc+
ac+
d ab ac ab cd
1 1
1

c

1

a
4

d

8
3
9

c
6

7
10
2
5

1a

b

b

a

c

bd+
cd+
ac+
ab abc

1
1

d
ab

1

ac

1

bc

1

Dim 0: [2,5], [3,6], [4,8], [1,inf]

Dim 1: [7.10], [9,inf]

Invariance▶ Persistent homology is a rigorous way of quantifying closed
shapes,▶ ... like connected components, cycles, voids and more.
▶ No matter if they are embedded in two or a million
dimensional space,▶ No matter if they are rotated, stretched or transformed in any
other way.
▶ multi-scale,▶ robust.

Lots of B, or a single A?

B BB
BB B BB B B B
B B
B B
B B
B B
B B
B BB BBBB B
B B
B B
B B
B B
B B
B B
B B
B B
B B
B B
B B
B B
B B

22

Lots of small circles, or a large one?

Multiscale▶ Persistent homology is a rigorous way of quantifying closed
shapes,▶ ... like connected components, cycles, voids and more.
▶ No matter if they are embedded in two or a million
dimensional space,▶ No matter if they are rotated, stretched or transformed in any
other way.
▶ Multi-scale,▶ robust.

Robustness
1

1.8

0.8

1.6

0.6

1.4

0.4

1.2

0.2

1

0

0.8

-0.2

0.6

-0.4

0.4

-0.6

0.2

-0.8
-1

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1.4
1.2

1

1

0.5

0.8
0

0.6
0.4

-0.5

-1

0.2
-1

-0.5

0

0.5

1

1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

2
1
1.5

0.9
0.8

1

0.7
0.5

0.6
0.5

0

0.4

-0.5
-1

0.3
0.2
-1

-0.5

0

0.5

1

1.5

2

0.2

23

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Robustness▶ Persistent homology is a rigorous way of quantifying closed
shapes,▶ ... like connected components, cycles, voids and more.
▶ No matter if they are embedded in two or a million
dimensional space,▶ No matter if they are rotated, stretched or transformed in any
other way.
▶ Multi-scale,▶ Robust.

Distances between diagrams

Optimal matchings between points of two persistence diagrams
allow us to define standard distances between them – bottleneck(length of the longest edge in the matching) and p-Wasserstein(sum of lengths of matching lines to the power q) to the power q1 .

Practical exercise 2▶ Let us go back to our jupyter-notebooks exercises.
▶ Open PH persistence simple point cloud,▶ Compute persistent homology of a point cloud sampled froma circle (without and with a considerable amount of noise).

24

Warning, outliers!

Outlayers can be a problem, filtration weighted by a distance tomeasure estimators

Warning, outliers!

outliers can be a problem, filtration weighted by a distance tomeasure estimators

Warning, outlayers!

outliers can be a problem, filtration weighted by a distance tomeasure estimators

25

Not only point clouds....

▶ If you work with:▶ Pixel / voxel / cubical data,▶ Time series,▶ Correlation and similarity measures,▶ ...

▶ you may still use similar ideas and track connectedcomponents and holes emerging and disappearing.

Apply to digital images

Left – osteoporotic, right – normal bone (vertebrae).
Not only density, but mostly structure is responsible forosteoporotic fractures.

What is a cubical persistence?
▶ Sub-level sets of a function.
▶ Cubes enter from lower to highest function/filtration value.
▶ We track changes in homology of sub-level sets.

26

Practical exercise 3▶ Digital images are partially-constant discretization offunctions.
▶ Let us go back to our exercises.
▶ Open PH distance from circle,▶ In this exercise we will construct a cubical approximation of afunction f : [−2, 2]2 → R. f (x, y ) is a distance from (x, y ) toa unit circle x 2 + y 2 = 1.
▶ Let us visualize it as an image, and let us compute persistenthomology of the corresponding cubical complex.

Persistence for time series analysis

S&P-500 and crashes

27Persistence–based descriptors of nanoporous materials

Lee, Bathel, Dłotko, Mossavit, Smit, Hess, Quantifying similarity of pore-geometry in nanoporous materials,
Nature Communications, 15396

And more...

▶ We do not have time to cover all this ground.
▶ But, there are numerous resources for further work:▶ https://arxiv.org/abs/1807.08607▶ https://www.maths.ed.ac.uk/~v1ranick/papers/edelcomp.pdf▶ https://gudhi.inria.fr/tutorials/▶ and many more...

Persistent homology▶ We have robust,▶ multi scale,▶ coordinate–free,▶ compressed,▶ tool to detect connected components, cycles, voids and theirgeneralizations.
▶ It can be interfaced in various ways with standard stat. and
ML tools.

28Persistent homology, the output
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

▶ Muti set of points in R2 .

▶ Variable size, not ideal representation to interface with ML/AI
and statistics → persistence representations, embeddings, ...
▶ We need to embed persistence diagrams into a Hilbert space(vectorize them).
▶ That makes topological/statistical inference - hypothesis
testing, confidence intervals,... possible.

Homology and persistent homology, biased collection of
resources▶ Edelsbrunner and Harer, Computational Topology, An
Introduction, AMS.
▶ Kaczynski, Mischaikow, Mrozek, Computational Topology,
Springer 2003.
▶ Dłotko, Applied and Computational Topology, Tutorial▶ Multiple youtube videos.

Persistence is nice, but, what about flares?

=

Persistence homology of those two point clouds will be very
similar, as they both have one connected component and one hole.

29But, what about flares?

=

But, oftentimes the information in the flares may be important (itmay for instance carry information about anomalies).

Reeb graphsource: Wikipedia

Reeb graph, formally▶ Input: M, f : M → R.
▶ We define an equivalence relation x R y iff:

▶ f (x) = f (y ),▶ x and y belong to the same connected component of f −1 (x).

▶ M/R .

30

Conventional Mapper algorithm

Conventional mapper graph is an attempt to define Reeb graph fordiscrete point cloud instead of a manifold.

Mapper algorithm, idea

▶ Input: finite collection of points sampled from M, f : M → R.
▶ We define a relation x R y iff:
▶ f (x) is close to f (y ),▶ x and y belong to the same cluster ...

Conventional Mapper algorithm

31

Conventional Mapper algorithm

Mapper algorithm, formally▶ Input: finite collection of points sampled from M, f : M → R.
▶ Cover of the range of f with overlapping boxes.
▶ Fix a clustering algorithm▶ We define a relation x R y iff:

▶ f (x) and f (y ) belong to the same element I of a cover of therange of f ,▶ x and y belong to the same cluster in f −1 (I ).

▶ Vertices of Mapper graph corresponds to the clusters,▶ An edge is placed between two vertices if the corresponding
clusters have nonempty intersection.

Mapper algorithm, coloring▶ Vertices of the Mapper graph may be colored by an averagevalue of an objective function on points covered by clusters.
▶ Fix a point cloud X and an objective function f : X → R.

▶ Each vertex of the Mapper graph correspond a subset(cluster) of points from X .
▶ Typically the value of the vertex will be an average value of fon the corresponding cluster.

32

Mapper is the most well known tool of TDA

Nicolau, Levine, Carlsson, Topology based data analysis identifies a
subgroup of breast cancers with a unique mutational profile and excellent
survival, PNAS 2011.

Practical exercise 1▶ Let us play with Mapper algorithm!
▶ Go to https://github.com/dioscuri-tda/tutorials▶ Let us start from something simple – open
Mapper concentric circles▶ In this exercise we will generate two concentric circles in aplane.
▶ We will use projection to the y coordinate as a lens function,▶ And a DBSCAN with certain parameters as a clustering
algorithm.
▶ What is the Mapper graph we obtain?

Practical exercise 2▶ Let us play with something more advanced, let us consider
standard Boston property dataset.
▶ Please open Mapper boston dataset▶ It contains 13 variables, we want to understand its relation to
prices of properties in Boston area (in ’1970).
▶ Here we will use t-distributed stochastic neighbor embedding
as a filtering function.
▶ We will be able to experiment with numerous clustering
methods as well.
▶ Obtained mapper graphs will be colored by the average priceof a property in a given cluster.
▶ This is not the last time we see Boston Property Dataset!

33

Ball Mapper algorithm▶ As the last part of our schedule, we will play with Ball Mapper
algorithm.
▶ As you might have noticed, it is not always trivial to choose
the lens function as well as clustering algorithm in standard
Mapper construction.
▶ The idea of Ball Mapper is intuitively explained in thefollowing slides.

Ball Mapper algorithm

Take a point cloud X

Ball Mapper algorithm

Given ϵ > 0, select subset of points N ⊂ X such that for everyx ∈ X there exists n ∈ N such that d(x, n) ≤ ϵ (we call N an
ϵ-net)

34

Ball Mapper algorithm

Consequently X ⊂n∈N B(n, ϵ), i.e. {B(n, ϵ), n ∈ N} cover X .

Ball Mapper algorithm

Take one dimensional nerve of that cover (an abstract graph whose
vertices correspond to B(n, ϵ), and edges to nonempty
intersections of balls)

Ball Mapper algorithm

This way we obtain a Ball Mapper graph of X with radius ϵ.
Vertices of the graph can be colored analogously to those ofstandard Mapper graph.

35

Network based landscapes of data

Meet the Lucky Cat

Network based landscapes of data

128 × 128 = 16384 dimensional space

From a gray scale image to a point1(p1,…,p16384)

3(p3,…,p16384
)
2

2
16384

(p ,…,p5(p5,…,p16384
)
4

)

4
16384

(p ,…,p)6(p6,…,p16384)

Gray scale images converted to vectors in high dimensional space36

Network based landscapes of data

128 × 128 = 16384 dimensional space

Practical exercise 1▶ Please open BM basic circle.
▶ In this proof-of-concept example we will generate a collectionof points sampled from a unit circle x 2 + y 2 = 1.
▶ And built a Ball Mapper graph based on it.
▶ Do we see what we expected to see?

Practical exercise 2▶ In our second example we will re-visit already known Boston
Property Dataset.
▶ Please open BM Boston property▶ This time we will use Ball Mapper to examine the structure of
the 13 dimensional point cloud, and the distribution of the
explanatory variable (price of properties) on the top of it.
▶ We will use tools from the Ball Mapper implementations torecognize which coordinates makes most statistical differences
between the regions of the graph.

37

Basic stats

Topology and hypotehesis testing

Basic stats▶ One-sample problem: We are given a data sample
X = {x1 , x2 , . . . , xn }, xi ∈ R d and cumulative distributionfunction F : R d → [0, 1]. Does the data X follow the
distribution F : X ∼ F ?
H0 : X ∼ F vs. H1 : X ≁ F▶ Two-sample problem: We are given two samples X1 ∼ F1
and X2 ∼ F2 and want to test hypothesis that X1 and X2 weredrawn from the same (unknown) distribution
H0 : F1 = F2 vs. H1 : F1 ̸= F2

Testing, for one-sample problem
Available methods depends on the data dimension▶ 1-D: plenty of available tests: e.g. Kolmogorov-Smirnov,
Cramer-von Mises, Andersonâ€“Darling, Chi-squared,
Shapiro-Wilks▶ 2-D: theoretical results for Kolmogorov-Smirnov andCramer-von Mises, some implementations available in python
and R▶ d-D: Kolmogorov-Smirnov should work but noimplementation available, critical values of test statistics
unknown, impractical in higher dimensions

Kolmogorov-Smirnov test
Here, K-S will be used as benchmark▶ one-sample: Dn = supx |Fn (x) − F (x)|▶ two-sample:
Dn,m = supx |F1,n (x) − F2,m (x)|38Testing, for one-sample problem
Available methods depends on the data dimension▶ 1-D: plenty of available tests: e.g. Kolmogorov-Smirnov,
Cramer-von Mises, Andersonâ€“Darling, Chi-squared,
Shapiro-Wilks▶ 2-D: theoretical results for Kolmogorov-Smirnov andCramer-von Mises, some implementations available in python
and R▶ d-D: Kolmogorov-Smirnov should work but noimplementation available, critical values of test statistics
unknown, impractical in higher dimensions

Kolmogorov-Smirnov test
Here, K-S will be used as benchmark▶ one-sample: Dn = supx |Fn (x) − F (x)|▶ two-sample:
Dn,m = supx |F1,n (x) − F2,m (x)|TopoTests, one-sample problem, input

We are given a data sample X = {x1 , x2 , . . . , xn }, xi ∈ R d andcumulative distribution function F : R d → [0, 1].

One sample TopoTests

Input: sample X = {x1 , x2 , . . . , xn }, xi ∈ R d and CDF
F : R d → [0, 1].
Step 1: EF (χ(n, r )), the Blueprint of F
′ from F▶ draw n-element samples X1′ , X2′ , . . . , XM
▶ for each sample Xi′ compute its ECC χ(Cr (Xi′ ))
▶

M

1 
a.s.
χ(Cr (Xi′ )) −−−−→ EF (χ(n, r ))
M→∞
M
i=1

39

One sample TopoTests

Input: sample X = {x1 , x2 , . . . , xn }, xi ∈ R d and CDF
F : R d → [0, 1].
Step 2: variation form EF (χ(n, r ))▶ draw a new set of m-element samples Y1′ , Y2′ , . . . , Ym′ from F▶ Calculate sup distance between χ(Cr (Yi′ )), i = 1, . . . , m andaverage ECC▶ determine the threshold value tα as a (1 − α)’th quantile of{di }mi=1 , where α is required level of statistical significanceTopoTests
Input: sample X = {x1 , x2 , . . . , xn }, xi ∈ R d and CDF
F : R d → [0, 1].
Step 3: Actual testing▶ compute the ECC for sample
data X : χ(Cr (X ))▶ compute the l∞ between
χ(Cr (X )) and EF (χ(n, r ))D = sup |χ(Cr (X ))−EF (χ(n, r ))|r ∈R▶ reject H0 if D > tα
▶ it is possible to get p-value as
well
For the two-sample problem theprocedure is slightly different but theidea remains.

TopoTests – properties

Design and goals▶ general method: works regardless of the data dimension andform of probability distribution function we are testing against▶ computationally feasible in higher dimensions▶ theoretical results derived (no ML-like approach)▶ in fact it is framework not one particular test
▶ outperforms baseline methods i.e. Kolmogorov-Smirnov test40

TopoTests – Translational & rotational invariance
The test is not sensitive to:▶ change of location parameter▶ rotation▶ reflection▶ components reordering (c.f. N × Γ vs. Γ × N )▶ This can be tested using standard moments (after topotest is
done)
6

y

4

N



(4, 4),



1.0 −0.8−0.8 1.0

2

0

N
−2
−2



0

(0, 0),

2x

1.0 0.8
0.8 1.0

4

6

Theoretical guarantees

Type II erorr (false negative, fail to reject H0 when it is false
For fixed significance level α, probability of type II error goes to 0
exponentially with number of points sampled
2

P(type II error) ≤∼ e −n → 0(Technical details swapped under the rug)Two-sample problem▶ X ∼ F , Y ∼ G , |X | = n, |Y | = m,

80
60
ECC

H0 : F = G vs. H1 : F ̸= G

100

20
0

▶ compute distance D between ECC curves on
X and Y0.0

▶ p-value is obtained as 
p = p I d(p) > D /N

0.2

r

0.3

0.4

0.5

40
30

▶ split Z randomly into X(p) and Y(p) of same
sizes

X(p)

20

Y(p)

10
0
0.0

0.1

0.2

r

0.3

0.4

0.5

50
40
30
ECC

▶ compute distance d(p) between ECC build on
X(p) and Y(p)

0.1

50

ECC

▶ data samples are pooled Z = X ∪ Y

X
Y

40

X(p)

20

Y(p)

10
0
0.0

41

0.1

0.2

r

0.3

0.4

0.5

Simulation results (one-sample)
Test Power: probability that H0 is correctly rejected when H1 istrue▶ samples sizes 100–5000 data points▶ test power estimated using 1000 MC replications▶ power compared with KS (d ≤ 3)

▶ α on diagonal is expected▶ distributions easy to confuse with normal:▶ t-Student with ν = {3, 5, 10} DoF▶ MVN non-diagonal Σ matrix▶ Cartesian products with N (0, 1) marginals▶ TopoTests yielded higher power than KS in most of the cases▶ Heavy MC simulations powered by Google.

Simulation results (one-sample)
KS 3D n=250 alpha=0.05

1.0

Average Power: 0.8087

N01xN01xN01 0.05 0.09 1.00 1.00 0.08 0.23 0.97 0.52 0.12 0.90 1.00 1.00 1.00 1.00
MultiGauss0.1 0.08 0.05 0.99 1.00 0.14 0.30 0.98 0.61 0.19 0.93 1.00 1.00 1.00 1.00
MultiGauss0.5 1.00 0.99 0.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.8

MultiGauss0.9 1.00 1.00 1.00 0.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

TRUE Distribution

N01xN01xT5 0.08 0.19 1.00 1.00 0.05 0.14 0.89 0.28 0.08 0.70 1.00 1.00 1.00 1.00
N01xT5xT5 0.20 0.29 1.00 1.00 0.08 0.05 0.56 0.11 0.05 0.30 1.00 1.00 1.00 1.00

0.6

T3xT3xT3 0.99 0.99 1.00 1.00 0.86 0.43 0.05 0.20 0.62 0.17 1.00 1.00 1.00 0.92
T5xT5xT5 0.52 0.60 1.00 1.00 0.20 0.08 0.24 0.05 0.12 0.14 1.00 1.00 1.00 1.00
T10xT10xT10 0.13 0.19 1.00 1.00 0.06 0.07 0.71 0.15 0.05 0.48 1.00 1.00 1.00 1.00

0.4

LaplacexLaplacexLaplace 0.92 0.95 1.00 1.00 0.62 0.24 0.24 0.09 0.37 0.05 1.00 1.00 1.00 1.00
LogisticxLogisticxLogistic

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.05 1.00 1.00 0.95

B22xB22xB22 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.05 1.00 1.00
UxUxU 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.05 1.00

0.2

ALTERNATIVE Distribution

GM1

UxUxU

B22xB22xB22

LogisticxLogisticxLogistic

LaplacexLaplacexLaplace

T5xT5xT5

T10xT10xT10

T3xT3xT3

N01xT5xT5N01xN01xT5

MultiGauss0.9

MultiGauss0.5

MultiGauss0.1

N01xN01xN01

GM1 1.00 1.00 1.00 1.00 1.00 1.00 0.89 1.00 1.00 1.00 0.96 1.00 1.00 0.05
0.0

Simulation results (one-sample)
TopoTest 3D n=250 alpha=0.05 norm=sup

1.0

Average Power: 0.9016

N01xN01xN01 0.05 0.05 0.85 1.00 0.30 0.86 1.00 0.99 0.60 1.00 1.00 1.00 1.00 1.00
MultiGauss0.1 0.04 0.06 0.84 1.00 0.34 0.88 1.00 1.00 0.63 1.00 1.00 1.00 1.00 1.00
MultiGauss0.5 0.83 0.77 0.04 1.00 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
MultiGauss0.9

0.8

1.00 1.00 1.00 0.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

N01xT5xT5 0.92 0.93 1.00 1.00 0.34 0.06 0.98 0.26 0.14 0.94 1.00 1.00 1.00 1.00

0.6

T3xT3xT3 1.00 1.00 1.00 1.00 1.00 0.99 0.03 0.77 1.00 0.14 1.00 1.00 1.00 0.94
T5xT5xT5 1.00 1.00 1.00 1.00 0.89 0.35 0.77 0.06 0.65 0.51 1.00 1.00 1.00 0.99
T10xT10xT10 0.67 0.70 1.00 1.00 0.12 0.08 1.00 0.53 0.04 0.98 1.00 1.00 1.00 1.00

0.4

LaplacexLaplacexLaplace 1.00 1.00 1.00 1.00 1.00 0.96 0.13 0.58 0.99 0.05 1.00 1.00 1.00 0.96
LogisticxLogisticxLogistic 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.06 1.00 1.00 1.00
B22xB22xB22 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.04 1.00 1.00
UxUxU 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.07 1.00

0.2

ALTERNATIVE Distribution

42

GM1

UxUxU

B22xB22xB22

LogisticxLogisticxLogistic

LaplacexLaplacexLaplace

T10xT10xT10

T5xT5xT5

T3xT3xT3

N01xT5xT5N01xN01xT5

MultiGauss0.9

MultiGauss0.5

MultiGauss0.1

GM1 1.00 1.00 1.00 1.00 1.00 1.00 0.98 1.00 1.00 0.98 1.00 1.00 1.00 0.06
N01xN01xN01

TRUE Distribution

N01xN01xT5 0.40 0.41 1.00 1.00 0.05 0.27 1.00 0.80 0.07 1.00 1.00 1.00 1.00 1.00

0.0

Simulation results (one-sample)
TopoTest 5D n=500 alpha=0.05 norm=sup

1.0

1.00

1.00

0.09

1.00

1.00

MultiGauss0.5

1.00

1.00

0.04

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

MultiGauss0.9

1.00

1.00

1.00

0.06

1.00

1.00

1.00

1.00

1.00

1.00

1.00

T3xT3xT3xT3xT3

1.00

1.00

1.00

1.00

0.05

0.99

1.00

1.00

1.00

0.17

1.00

T5xT5xT5xT5xT5

1.00

1.00

1.00

1.00

0.96

0.05

0.87

0.12

1.00

0.78

0.05

T10xT10xT10xT10xT10

0.89

0.94

1.00

1.00

1.00

0.77

0.06

0.29

0.34

1.00

0.64

N01xT5xT5xT4xT5

1.00

1.00

1.00

1.00

1.00

0.12

0.38

0.06

0.97

0.99

0.12

N01xN01xN01xN01xT5

0.17

0.22

1.00

1.00

1.00

1.00

0.23

0.93

0.07

1.00

0.99

LapxLapxLapxLapxLap

1.00

1.00

1.00

1.00

0.22

0.91

1.00

1.00

1.00

0.04

0.96

N01xN01xLapxLapxLap

1.00

1.00

1.00

1.00

1.00

0.12

0.85

0.22

1.00

0.91

0.07

LapxLapxLapxLapxLap

0.14

1.00

N01xN01xLapxLapxLap

1.00

0.83

N01xT5xT5xT4xT5

0.82

1.00

N01xN01xN01xN01xT5

1.00

1.00

T5xT5xT5xT5xT5

1.00

1.00

T10xT10xT10xT10xT10

1.00

1.00

MultiGauss0.9

1.00

0.04

T3xT3xT3xT3xT3

0.08

0.05

MultiGauss0.5

0.06

MultiGauss0.1

MultiGauss0.1

N01xN01xN01xN01xN01

N01xN01xN01xN01xN01

TRUE Distribution

Average Power: 0.8465

0.8

0.6

0.4

0.2

0.0

ALTERNATIVE Distribution

Simulation results (one-sample)
Average power, dim=2

Average power, dim=1
0.95

0.9
Average power

Average power

0.90
0.85
0.80

0.8

0.7

0.75
0.70
0.65

0.6

TopoTest
KS
0

1000

2000
3000
Sample size, n

4000

5000

2000
3000
Sample size, n

4000

5000

4000

5000

Average power, dim=5
TopoTest
KS

0.95

0.95

0.90

0.90

Average power

Average power

1000

Average power, dim=3

1.00

0.85
0.80

0.85
0.80
0.75
0.70

0.75
0.70

TopoTest
KS
0

TopoTest
KS
0

1000

2000
3000
Sample size, n

4000

0.65

5000

0

1000

2000
3000
Sample size, nSimulation results (two-sample)
1.00

0.69

1.00

MultiGauss0.9

1.00

1.00

1.00

0.06

0.98

1.00

1.00

1.00

1.00

1.00

1.00

T3xT3

0.98

0.99

0.99

0.98

0.06

0.33

0.84

0.88

0.09

0.82

0.74

T5xT5

0.80

0.81

0.97

1.00

0.36

0.05

0.27

1.00

0.23

0.26

0.85

T10xT10

0.28

0.31

0.67

1.00

0.83

0.24

0.05

1.00

0.72

0.05

0.99

LogisticxLogistic

1.00

1.00

1.00

1.00

0.89

1.00

1.00

0.06

0.99

1.00

0.96

LaplacexLaplace

0.98

0.98

1.00

1.00

0.09

0.21

0.75

0.99

0.06

0.73

0.76

N01xT5

0.34

0.30

0.72

1.00

0.83

0.26

0.06

1.00

0.72

0.06

1.00

GM_1xGM_1

1.00

1.00

1.00

1.00

0.71

0.86

0.99

0.97

0.75

1.00

0.06

0.6

0.4

ALTERNATIVE Distribution

1.0

2-sample KS 2D n=250 alpha=0.05 norm=sup
Average Power: 0.6030

0.8

0.2

0.0

▶ TopoTests still outperforms
the KS (d = 2, n = 250,
0.765 vs. 0.603)▶ very expensive method

N01xN01

0.05

0.04

0.55

1.00

0.33

0.07

0.03

1.00

0.17

0.05

1.00

MultiGauss0.1

0.05

0.05

0.36

1.00

0.35

0.15

0.08

1.00

0.22

0.08

1.00

MultiGauss0.5

0.73

0.46

0.05

0.80

0.88

0.78

0.73

1.00

0.80

0.69

1.00

MultiGauss0.9

1.00

1.00

0.86

0.05

1.00

1.00

1.00

1.00

1.00

1.00

1.00

T3xT3

0.44

0.40

0.83

1.00

0.05

0.09

0.16

0.90

0.09

0.14

0.74

T5xT5

0.10

0.17

0.71

1.00

0.09

0.05

0.08

0.99

0.10

0.08

0.93

T10xT10

0.05

0.08

0.64

1.00

0.15

0.07

0.05

1.00

0.07

0.06

0.99

LogisticxLogistic

1.00

1.00

1.00

1.00

0.92

1.00

1.00

0.05

0.98

1.00

0.26

LaplacexLaplace

0.30

0.30

0.80

1.00

0.12

0.13

0.11

0.98

0.06

0.15

0.94

N01xT5

0.07

0.09

0.60

1.00

0.13

0.07

0.06

1.00

0.10

0.05

0.99

GM_1xGM_1

1.00

1.00

1.00

1.00

0.74

0.92

0.99

0.24

0.93

0.99

0.05

N01xT50.30

1.00

GM_1xGM_1

0.98

1.00

LaplacexLaplace

1.00

0.67

T10xT10

0.29

0.96

LogisticxLogistic

0.81

0.99

T5xT5

0.99

1.00

T3xT3

1.00

0.06

MultiGauss0.9

0.21

0.22

N01xN01

0.06

0.24

MultiGauss0.5

0.05

MultiGauss0.5

MultiGauss0.1

MultiGauss0.1

1.0

TRUE Distribution

1.00

N01xT50.28

GM_1xGM_1

0.98

LaplacexLaplace

1.00

T5xT5

0.32

LogisticxLogistic

0.81

T3xT3

0.98

T10xT10

1.00

MultiGauss0.9

0.22

MultiGauss0.5

0.06

N01xN01

0.05

MultiGauss0.1

Average power, dim=2
TopoTest
KS

0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
100

43

0.8

0.6

0.4

ALTERNATIVE Distribution

Average power

TRUE Distribution

2-sample TopoTest 2D n=250 alpha=0.05 norm=sup
Average Power: 0.7653

N01xN01

150

200

250
300
350
Sample size, n

400

450

500

0.2

0.0

TopoTests, take home message▶ There are multiple papers where topological techniques areused to show differences in distributions▶ Usually they work
▶ We shown an important case, where it works, is comparable orbetter than state of the art in low dimension and have nocompetitions in high dimensions▶ Not only that, we have theoretical guarantee for that
▶ Those guarantees does not depend on the fact that we started
from point clouds▶ We hope that this meta–observation will open up newopportunities in applied topology

Every mathematician has a secret weapon. Mine isMorse theory.
Raoul Bott

Discrete Morse Theory

1. Let us now have a look at a Discrete Morse Theory.
2. K - finite regular CW complex.
3. f : K → R, constant on every cell, is a discrete Morsefunction if for every αp ∈ K:
3.1 #{β p+1 > αp |f (β p+1 ) ≤ f (αp )} ≤ 1
3.2 #{γ p−1 < αp |f (γ p−1 ) ≥ f (αp )} ≤ 1

4. Simplex is critical if both (1) = 0 and (2) = 0.
5. For any simplex conditions (1) and (2) cannot be both = 1(=⇒ define discrete gradient).

44

Which of them is discrete Morse function?

Why it is called discrete Morse Theory
1. Suppose K is a cell complex with a discrete Morse function.
Then K is homotopy equivalent to a CW complex with exactly
one cell of dimension p for each critical simplex of dimensionp (we will construct this complex soon).
2. If there are no critical simplices a with f (a) ∈ (a, b], thenK(b) is homotopy equivalent to K(a). (In fact, K(b) collapses
to K(a)).

3. If there is a single critical simplex a with f (a) ∈ (a, b] thenK(b) is homotopy equivalent to K(a) with a handle ofdimension dim(a) glued.
4. Morse inequities hold.
5. Gradient of a function is more convenient to use then afunction itself.

Discrete Gradient45

How discrete Morse functions are usually constructed?

1. We almost never assign the values. Gradient is sufficient.
2. It will be represented by arrows.
3. Every simplex can be either tail of head of exactly one arrow.
4. The vector field is curl-free (i.e. there are no loops).
5. Critical cells of Morse functions = cells which are unpaired.

Illustration

Illustration46

Illustration

Illustration

Illustration

47

Illustration

Illustration

Illustration48

Illustration

Illustration

Illustration

49

The Morse complex▶ Cells of Morse complex = critical cells of discrete vector field.
▶ Boundary relation computed by using gradient paths.
▶ Over Z2 – κ(A, h) =number of gradient paths from A to h
mod 2.
▶ Morse complex (over integers) and the initial complex are
homotopically equivalent.
▶ Homology of a complex and its Morse complex - isomorphic.
▶ κ(A, h) = 0.

A

b

d

c

e

g

f

h

Iterated Morse Complex▶ By iterating construction of a Morse complex we can obtain
both (field) homology and persistence.
▶ Let us concentrate first on standard homology.
▶ Homology over a field =⇒ pairing between A, B can be madeiff κ(A, B) ̸= 0 (Dmitry Kozlov).
▶ Algorithm to construct Morse complex – a functor
M : C → C.
▶ C category of chain complexes.

▶ Assumption: if there are some Morse pairings in C , at least
one of them is made in M(C ) (vitality).
▶ E.g. M procedure search for a single possible pairing and do it.

Iterated Morse Complex and homology▶ Let us apply M iteratively.

▶ Homology is preserved, homotopy type is not.
▶ ∃n∈N Mn (C ) = Mn+1 (C ) = . . . =: M∞ (C ) – Iterated Morsecomplex.
▶ βi (C ) = #{ cells in M∞ (C ) of dimension i}.

▶ Generators can be obtained from this procedure.

50

The Dounce hat.
0

2

2

1

1

1

0

2

0

First iteration pairings
0

2

2

1

1

1

0

2

0

Boundary on the first iteration
0

2

2

1

1

0

1

2

51

0

M1 (K) with Morse pairings on it

Morse complex for persistence▶ C – chain complex with filtration g : C → Z.

▶ s.t. a, B ∈ C , a < B =⇒ g (a) ≤ g (B).
▶ Morse pairing v : C → C is compatible with filtration if
g (a) = g (v (a)) for every paired a.
▶ Assumption: M constructs only a vector fields compatible
with filtration.
▶ Persistence of C and M(C ) are the same.

Filtered complex

0

1

0

1

0

1

0

1

0

1

0

1

0

1

52

Filtered complex

0

2

2

2

2

1

0

1

0

1

0

1

0

1

0

1

0

2

2

2

1

2

2

2

2

2

1

Filtered complex

0
0

1

0

1

0

1

0

1

0

1

0

2

3

2

2

1

2

2

2

4

2

2

1

Filtered complex

0
0

1

0

1

0

1

0

1

0

1

0

2

2

3

53

2

2

1

Filtered complex

0

2

2

2

4

2

0

1
5

5

0

5

0

5

0

5

5

5

5

5

5

1

5

5

5

1

5

5

0
0

1

1

1
2

3

2

2

1

2

2

2

4

2

2

1

6

6

6

6

6

1

5

5

5

5

5

0

5

5

5

5

5

1

0

5

5

5

5

1

5

0

6

6

6

6

6

1

0

2

3

2

2

1

2

Filtered complex000Iterated Morse Complex for persistence▶ Dimension 0 – [0, ∞), [1, 3].
▶ Dimension 1 – [4, 6], [5, 6].

541

First iteration
4

2

2

2

6

6

6

5

5

0

5

0

2

1

6

6

1

5

5

5

5

5

5

5

1

5

5

5

5

1

5

0

6

6

6

6

6

1

0

2

3

2

2

1

2

2

2

4

2

2

1

6

6

6

6

6

1

5

5

5

5

5

0

5

5

5

5

5

1

0

5

5

5

5

1

5

0

6

6

6

6

6

1

0

2

3

2

2

1

2

2

2

4

2

2

1

6

6

6

6

6

1

5

5

5

5

5

0

5

5

5

5

5

1

0

5

5

5

5

1

5

0

6

6

6

6

6

1

0

2

3

2

2

1

2

0
0
0

1

Second iteration

0
0
0

1

Final iteration

0
0
0

55

1

Critical cells
2

2

0

2

4

2
1

0

1

6

6

6

6

6

5

5

5

5

5

0

5

5

5

5

5

1

0

5

5

5

5

1

5

0

6

6

6

6

6

1

0

2

2

3

2

2

0

1

1

Critical cells

a

b

4
c

0

1
Y
6

(Y,6)

(X,6)

5
d
(c,4)

X
6

(d,5)

(e,3)

(a,0)

(b,1)

(Y,6)

(X,6)

e
3

Critical cells

c
X
a

d

b

(c,4)

(d,5)

(e,3)

Y
(a,0)
e

56

(b,1)

Observations▶ A ∈ M ∞ (C ), and B1 , . . . , Bn be in boundary of A in M ∞ (C ).
▶ g (A) > g (B1 ), . . . , g (Bn ).
▶ M ∞ (C ) is the minimal cell complex (w.r.t number of cells)with the same persistence as C .

Observations(Y,6)

Dim 0:
[1,3]
[1,∞)

(X,6)

(d,5)

(c,4)

(e,3)

(a,1)

(b,1)

(Y,6)

(X,6)

Observations

Dim 1:[4,6][5,6]

Dim 0:
[1,3]
[1,∞)

(d,5)

(c,4)

(a,1)

57

(e,3)

(b,1)

Persistent Homology via DMT▶ Based on Morse theory one can obtain persistent intervals.
▶ No need to change representation for one suitable for matrix
algorithm.
▶ Unlike the simplification phase, cells of different filtration
value are paired and nonzero persistent intervals are reported.
▶ Pairings between cells of different filtration value – allowed (tosome extent).

Level 0

c
(Y,6)

(X,6)

Y
d

a

b

(c,4)

(d,5)

(e,3)

X
(a,1)

(b,1)

(Y,6)

(X,6)

e

Level 3

c
Y
d

a

b

(c,4)

(d,5)

(e,3)

X
(a,1)
e
Dim 0:

[1,3]

58

(b,1)

Level 3
c
(Y,6)

(X,6)

Y
d

a

(c,4)

(d,5)

X
(a,1)
Dim 0:[1,3]

Level 6
c
(Y,6)

(X,6)

Y
d

a

(c,4)

(d,5)

X
(a,1)
Dim 0:

Dim 1:

[1,3]

[5,6]

Level 6
c
(X,6)

X

a

(c,4)

(a,1)
Dim 0:

Dim 1:

[1,3]

[5,6]

59

Level 6
c
(X,6)

X

a

(c,4)

(a,1)
Dim 0:

Dim 1:[1,3][5,6], [4,6]Level ∞

a(a,1)
Dim 0:

Dim 1:[1,3], [1,∞) [5,6], [4,6]The story begins here?

1. We have barely scratch the surface,
2. there are many more invariants,
3. and more applications.
4. Hence, I would like to invite you to Topological Data Analysis!

60Collaborators

This research has been conducted over several years with theinvaluable contributions of numerous great collaborators:
1. Phd students: Ahmad Farhad, Davide Gurnari, Niklas
Hellmer, Jakub Malinowski, Jan Senge,
2. Postdocs: John Harvey, Tak-Shing Chan (Swansea), Michal
Lipinski, Bartosz Naskrecki, Justyna Signerska-Rynkowska,
Anastasios Stefanou, Rafal Topolnicki (Dioscuri),
3. Collaborators: Senja Barthel, Hubert Wagner, Simon Rudkin,Radmila Sazdanovic, Berend Smit, Alex Smith, Ruben
Specogna, Lukasz Stettner and others

SupportI gratefully acknowledge support by Dioscuri program initiated by
the Max Planck Society, jointly managed with the National ScienceCentre (Poland), and mutually funded by the Polish Ministry of
Science and Higher Education and the German Federal Ministry of
Education and Research.

Thank you for your time!
Dioscuri Centre in Topological Data Analysis@Facebook

Paweł Dłotko
pdlotko @ impan.pl
pdlotko @ gmail
pawel dlotko @ skype

61WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Medical imaging signatures with topology forcancer
Hidetaka Arimura
Faculty of Medical Sciences, Kyushu University, Japan
What the author is interested in is the connection between medicine and mathematics.
A human body is equivalent to a tube or donut (without considering holes of noseand eyes). The central hole is a digestive system. The body is covered by surface
tissue (epithelial cells). The epithelial cells exposed to the outside world might havegene mutations, thereby resulting in cancer cells. On the other side, the heterogeneityof pixel values in medical images (computed tomography, magnetic resonance imaging, positron emission tomography, etc) would reflect biological tumor heterogeneity,
which could be related to the degree of malignancy and patients’ prognoses. We haveattempted to develop novel medical imaging signatures, which are defined as sets offeatures calculated based on mathematical models from medical images, for predictionof the degree of malignancy and patients’ prognoses. As results, the author’s grouphas shown several data that the topological imaging signatures could be superior toconventional ones in terms of the prediction. The topological image features are derived from Betti number maps (b0 , b1 , and b2 ) within cancer regions of medical images.
The assumption that the author has thought through (not twisting things around) isthat the b0 , b1 , b2 features may characterize high tumor cell density areas, scattereddead cell areas (necrotic tissues), cancer blood vessels (angiogenesis), respectively. The
author will present the basics of topological image features and the applications to lungcancer and head and neck cancer.

63

Workshop on mathematics for industry
25-29 September 2023 (Warsaw)
Basis of mathematics in nanomedicine structures and life sensing

Hidetaka Arimura, PhD
Professor (medical Physics)
Division of Medical Quantum Science
Department of Health Sciences, Faculty of Medical Sciences
Kyushu University

I am a medical physicist, not amathematician

2!"#$%&&'()*&+,#!!"#$%&'()*+,-./%'./%+'/0,10("
2./$%34/0('5%6'()*+,-./%7,+%#,8,5,*0('5%+'/0,10("
9+/$%:8850('#0,."%,7%#,8,5,*0('5%+'/0,10("!

64

Outlineü Nature of medical physics
ü Association in shape between human body and topology
ü What we are doing now (radiomics)
ü Mathematical models beyond conventional radiomics
ü Summary (1)4Nature of medical physics

5

What is mathematics?
ü Mathematics is the art of giving the same
name to different things (Henri Poincar´e)

Abstract science︖

ü Mathematics is the structure of abstract
reasoning (Richard Philips Feynman)

665

What is physics?
ü Basic science that understands and describes
concrete natural phenomena by using

Concrete science︖

mathematics that can explain them
ü Basically, the natural phenomena could betheoretically predicted in the macroscopic world,but probabilistically predicted in the microscopic
world (quantum mechanics).

7

What is medicine?
ü Science of uncertainty and an art ofUncertain science︖

probability [William Osler (1849-1919) ,Principle and Practice of Medicine]ü Inherent uncertainty in health care [The
Lancet 2010; 375: 1666]

8What is medical physics (my field)?
Applied science that could describe natural phenomena

Concrete, but
Uncertain science?
with abstract spice︖?

related to human bodies with uncertainties (due tothermal motion or dynamic metabolic activity?) using
mathematics that can be used for diagnosis and therapy

966

What we can get in cancer properties

Big pictures on human body and diseases,
because you can only predict softly
something with uncertainties

フッターを追加

10“What is essential is visible to the heart.
It is invisible to the eye.” (modified from an original
version)(By a fox in “The Little Prince”)

11“The truth is invisible to the eye.
However, mathematics can reveal andexpress its appearance.”(By a mathematician in a novel of “The Doctor Loves
Equations” written by Youko Ogawa)

1267

“The book of theuniverse is written inmathematics”(Galileo Galilei）

Association in shape between humanbody and topology

17

What is a human body?

An embryoA grown set of cells

A small dent becomesan initial anus

Initial mouth

Initial anus
18

68What is a human body basically?
Our bodies are covered bysurface tissue (epithelial cells).

Mouth
A digestivesystem≈Anus≈

Epithelial cells of the digestivesystem exposed to the outside
world
They might be cancerous due togene mutations such as coloncancer.
20>%-*'4#.'10"-,-.%$?"#"+1))1<"+/"$,-01"'
234$3"&',$'56'.74%"8!"-4<,=.-"

12%34'-%*-(5'--/(#&(-8+,>(94%54("*6(;'('72#/'$(+&,-".$

3#(34'(/"#?'(#)(
54'"%5*-(34%+,/<(!"##"$%"&'()*:4'6("%,43(;'(-8+,(
5*+5')<+/"$,-01"'
234$3"&',$'+>6'.74%"8

9"$"':.-/'
7;-4-.,$!"#$%$&#'%(")'*++,-..")$/012,.34"5+6")$/0!"#$%$&#'%(")'!"#$%&'

What we are doing now (radiomics)

69)'

>(/'%$#'0%,0#$'2%*+#,*3'*$#%*#;=-'3%"31/'9%$#1"$;*/.1/5<+1()$+1/%95:%7#)$+1/%95'<,<("#-'58-*)@3*),'3'$($)8,/(&#)("83*3%#+/!((*")$+1/%95

A'.'-#2'$(;6(*(B*2*+'/'(C#.'-(-*8)'*3'(!DEFG0>(H)#&<(I#+J6#)#

How to decide eachtreatment approach : NSCLC

If a patient is operable with a stage IA tumor,









this patient will receive surgery,but if not, this patient could receive 

radiotherapy.
   Radiotherapy 

 

                

National Cancer Research Center, Cancer
Information Service
https://ganjoho.jp/public/cancer/lung/treatme
nt.html


 
27

4.5),)/6&5)&#$,)&$*&'((7)8&9$%')*%(&5'%#&(':'+$/&9/.;'+)(&%/)$%)<&-2&/$<'.%#)/$92&:$2&#$,)&<';;)/)*%&(7/,',$+&%':)(&!"#"$%&%"'()&*"#"$%&%"'(

7889.:$;<*.=8)>.=&)08=%11;$
03.%0?.*0$@3'4.A93.8$
4'91A$B.$9*89)0%B1.$&'3$
8'?.$@%0).*08C$=1"71/

6%&1>1(%&1?#,$)&).5

;<<

;<<

-.1

@A

@B

C81/%&&',$%.1

D'EFAGH6HI

D'EFAGH6HI;*/8#8%&'$#(1'E7%5I

AJKL

MJL

)%

70

!"#$%&'()%*)("+,'+-)./0)+.01%#2'+.3-).+').4&%(#)#2')(.&'5)6"#)#2'-).+')/%#)3'+*'$#!"#$%&'(()*+((,(-#.,*'#*+/,
0!1232415'.+,6=+%/4%>('3(*-<(M+3(B(L*$%*3(=+5#-(R%#-(H46/<(DEFF

D$03.%0?.*0$?.0-'A$-%8$B..*$.&&.40)E.$&'3$8'?.$@%0).*08F$B90$0-.$?.0-'A$)8$*'0$%1G%#8$.&&.40)E.$&'3$%11$@%0).*08C$)&@,"&%5'0'.'10"15"4#&'1$'(L*$%#"%5/O(#+'(#&(#"%5/()'/'*)54(&%'-$/(94')'(*(/'3(#&("'$%5*-(%"*,'(&'*38)'/()'-*3'$(3#(2*3%'+3/S(2)#,+#/'/(*)'(
5#+/%$')'$(T)*$%#"'U(-%?'(,'+#"'>(3)*+/5)%23#"'>(2)#3'#"'>("'3*;#-#"'>(3#(%"2)#.'($%*,+#/%/>(3)'*3"'+3>(*+$(
2)#,+#/%/-99&#3%$#)"O(N'-'53%#+(#&(2*3%'+3/(&#)(*("#)'(*22)#2)%*3'(3)'*3"'+3(/3)*3',6(;6(2)'$%53%#+(#&(#&(2*3%'+3/S(
2)#,+#/'/(#;3*%+'$(&)#"(2)'@3)'*3"'+3("'$%5*-(%"*,'/(!'75'23("#-'58-*)-6(3*),'3'$($)8,/(&#)(38"#)/(9%34(,'+'("83*3%#+/0((

7$.&4,$)$8&,.%&'"(<:/(3-*#9&%'()*%

2.&'$(&)
1&,.&'*'%.&'"(

94'#:'(0)"*)1'0(.&4,$1

6#4)#78./%9:08.3*.
#)(39#(%;..
/%-9%4(#(:34!"#$%&"'(")"*$+,+%,-,.(%/(0*%(%/(.
1.23'4%(5!"#$%!"#$%&'()*
!"#$%$&'()*&+,-"./0/
1$2&3-$4

34,5'5.#).(.#61'1)*",)
14,0$,6%&'("()*+,'-.)."*"/0!"##$%&'"()"*)+,$&,$.&/$(&)'/.0$1+,-.("*/

0"#1.("*/!"#$!(

614A5)1<"15"4#&'1$'(-"<'.+":!
7$.&4,$)$8&,.%&'"(
6#4)#78./%9:08.3*.
#)(39#(%;..
/%-9%4(#(:34<:/(3-*#9&%'()*%!"#$%

94'#:'(0)"*)1'0(.&4,$1

2.&'$(&)1&,.&'*'%.&'"()
41'(0)1'/+#$)/":$#1)",)/.%<'($)#$.,('(0!"#$%&"'(")"*$+,+%,-,.(%/(0*%(%/(.
1.=>!!?.3*.
23'4%(5>5.#4.&'"(!"#$%&$'()*#)*%%$"+"#$%&'&()(*+!"##$%&'"()"*)+,$&,$.&/$(&)'/.0$1);'&<)&.,0$&)+,"0("1'1=)&<.&)
6"4);.(&)&")+,$:'%&)!

!"#$%&'()*!"#$%&'#()*&'+,-''+.,+*&/+01(01(#(*&.2'/13&0#((-+&
024#,#05*&+0,6&

!"#$%$&'()*&+,-"./0/
1$2&3-$4!"#$%&'()*($(+),"-$./(0'10!)

71

7+'01$#1%/)%*)$./$'+)3+%,/%(1(8)9':3'$#'0)%"#$%&';).*#'+)#+'.#&'/#)*+%&)&'01$.4)1&.,'(!!

!!

,%$-(#&(."/)0%.1%$**(." 23$4&)&.)%(15&6)'"-)0'%&('3)%$*0."*$)2%(15&)&.)3$4&6
VH)#,+#/%/O('72'53'$(#835#"'(#)(#83-##?()'-*3'$(3#(*("'$%5*-(5#+$%3%#+(#)(3)'*3"'+3!!

<.#2'&.#1$.4)&%0'4()6'-%/0)$%/='/#1%/.4)+.01%&1$(%">2'+').+')01(.0=./#.,'()1/)$%/='/#1%/.4)+.01%&1$()./0)0''3)4'.+/1/,
H%A)'?)48$
  !
   -*$)(%$#3%&&5'&1%/"17'21%$*/1,'*,#".'7119'&1%/"#".'ENOI

   

! M+3')/5*++')(.*)%*;%-%36! W8-+')*;%-%36(&#)(.*)%#8/(%"*,%+,(
2)#3#5#-/! " # $% &N#,%78%"$%.1')2'NOO


"

  
    #

N#,%78%"$%.1')2'3)"81"$#)"%&'/%7#)(#,'21%$*/1,O!

! C'5'//%36(#&(48,'(+8";')(#&($*3*

D90'?%0.A$&.%093.$.(03%40)'*!#72

A#6&%6'$%'$B()'$5$+&*%/C%-$&&$'%C$6&('$*%)+%'68)/5)7*%
C'/5%5$8)76.%:/)+&%/C%,)$3D%! P1'*4%74'#-+4"%"<,-5/%+4754(#%,.4%,+%",14%,7%('.(4+%       


8<4.,#=84"!"#$%&'(%#)*+'%! P1'*4%74'#-+4"%"<,-5/%64% 
  

1'#<41'#0('55=%0.B'+0'.# 
       

!"#$ &'(%)*+,-++./01%2*+3/4+,5%6#71+*5+8/+9:#;<=>?5@#=25*+3/4+,!&

A#6&%6'$%'$B()'$5$+&*%/C%-$&&$'%C$6&('$*%)+%'68)/5)7*%C'/5%EF%
G:6&&$'+%'$7/H+)&)/+I%:/)+&%/C%,)$3D%! H*&$0%*#(#4.0;I("*(#)9);&%($.($4.#*(<.%(."J*'$#()+(&(#&9*('&$*F.%8/("0$($4*8(#4.0;I("*(I)<<*%*+$(<%.9($4.#*(<.%(."J*'$#()+(I)<<*%*+$('&$*F.%)*#(K?0I&(LMMMNO(! H*&$0%*#(#4.0;I("*()+:&%)&+$($.()%%*;*:&+$($%&+#<.%9&$).+#(.<($4*()+50$#(#0'4(&#($%&+#;&$).+/(%.$&$).+/(&+I(#'&;*O(='#$#.13
I''A$&.%093.8

7'1.(310+8&
1.8&'2010+8"%A$&.%093.8

JK

92:.(1/;3+8

JK<;(1/;3+8

JK
JK!"#$%&'()(*&+$,-%&.(/(*&0-1,2%&!(3(&.$--4,5&67$889:96$-915%&;5#&4#(*&<974=>?5-4,8694564@&A4B&C1,2%&AC%&D0E%&;FFF("(

B1*1)1=,"'-"10%"15".+%14'%-".+#."<%"<#0.
7.0.3.1(*&* #'")$'&)')#.44$$)#80)9(&5)&%:("1)&.)-%(";)#.44$$)4%.<)-."8&=)P*((5Q75$:)#'")".&)-(44$%$"&('&$))')#.44$$)#80)'"-)-."8&>"'73Hypothesis: Classification of cancer patterns into several categories
depending on Betti numbers
q Mathematical classification of objects by simplifying connectivity[G. Rote, in: Eff. Comput. Geom. Curves Surfaces, 2006: 277–312.]

q Betti number: Invariant value

Cancerpattern C!

Betti numbers after continuous
deformationb0: Number of connected components
b1: Number of holes

q b0 = 1
q b1 = 0

Cancerpattern A

q b0 = 1
q b1 = 1

Cancer patterns with prognostic information could be classified into several
categories based on Betti numbers with intrinsic geometrical patterns
42

Computational topology

Decomposition of a shape: a set of simplices (many triangles)= Vertices (Points) + Edges (Lines) + Faces

Vertex(point)Side(line)

Face

フッターを追加

43

C1$*3.#.'10#)".1*1)1=,?*'.95.#)$).+(.<(&(#4&5*P(&(#*$(.<(#)95;)'*#(69&+8($%)&+F;*#G(
Q(R*%$)'*#(6S.)+$#GT(AIF*#(6@)+*#GT(H&'*#&'(')'*+#*,'-./#!! " #$! # %&'( !! ""#$%!
&'%!"## %&'( ()%*! + %&'( ,-*!()%&'( ()%*! /#0123(4053'06)#7+7)45%&'( ,-*!() /#0123(4053'06)#!'-026,345

74!"#$%&'""

D3$$#4,"8E9! ?$)&%:)&.)4("-)*(1"(4(#'"&)(<'1$)4$'&8%$*)&.)#5'%'#&$%(@$)("&%("*(#)#'"#$%)1$.<$&%:! ?$)'**8<$)&5'&)&5$)*(1"(4(#'"&)(<'1$)4$'&8%$*)<':)A$)<'&5$<'&(#'33:)("B'%('"&>! ?$)A$3($B$)&5'&)&5$)(<'1$)4$'&8%$*)#.83-)A$)#'3#83'&$-)4%.<)&.0.3.1:-"5'R*1,$#)"')/'
3)((1"$,'S"#

75

Workshop on mathematics for industry
25-29 September 2023 (Warsaw)
Basis of mathematics in nanomedicine structures and life sensing

Hidetaka Arimura, PhD
Professor (medical Physics)
Division of Medical Quantum Science
Department of Health Sciences, Faculty of Medical Sciences
Kyushu University!"#$%&&'()*&+,#!"#$%&'()*+,-./%'./%+'/0,10("! 2./$%34/0('5%6'()*+,-./%7,+%#,8,5,*0('5%+'/0,10("%%%9+/$%:8850('#0,."%,7%#,8,5,*0('5%+'/0,10("!

!)*&+,#!:+4%#;4+4%+45'#0,.";08"%'1,.*%('.(4+%8+,84+#04"<%('.(4+%*4,14#+=%'./%8+,*.,"0">%!?,54%'.'5="0"%@#,8,5,*0('5%+'/0,10("A%'"%'.%4B85'0.'654%:C!D-11'+=%@2A"76

-(&#'%,%&./+/'0*(1(&(2+3%&'$%4+(5+3/6'%/'%,'#71&%+,%8&#'9:!"!""#$%&'%#()*+',++()*%#-#.%$&-)$#/0#(+('")#1)$&($+2)&(3)'#0#-#.4!"#$%&'()&*+"),(

3$.'.+$%&'(%.0/.1"1),($1(4$,).'.+$%&'($0&+",-./.'.+$%&'(%.0/.1"1),($1(
2-($0&+",! !"#$%&'()*! +-221! +(%%-,/-)&! +$%,-.&! 7-,.(161,(#0(%-%/1! !(2-1&! 82$%)&)",/&

3/"'-14! 9-11-2! +$56/6-1#$!"#$%$&$'%('#()'%%")&"*()'+,'%"%&-(.%*(/.)012'3%*(
4$&5$%(.(0"2%"6
E$:%$*(>"F$6(21*(&(51::$5%$#(51+>1:$:%(":(B(:$"C/D1*/11#
G$"C/D1*":C(>"F$6(21*(&(51::$5%$#(51+>1:$:%(":(B(:$"C/D1*/11#
B(:$"C/D1*/11#

H(:$"C/D1*/11#

E$:%$*(>"F$6(21*(D&5'C*1,:#(":(H(:$"C/D1*/11#
G$"C/D1*":C(>"F$6(21*(D&5'C*1,:#(":(H(:$"C/D1*/11#

!
"
#
$
%
&
'
(
):(1*"C":&6("+&C$I":&*7("+&C$(1D%&":$#(D7(&(%/*$3/16#(9&6,$(12(J77

0$2":"%"1:(12(51::$5%$#(51+>1:$:%3(4K/"%$8(&:#(
D&5'C*1,:#(4D6&5'8#%

5+1%(%'%#(")#1)$#((+$'+3)$#/0#(+('")&(3)6#-+"),%'6%()&)7+2(+!"#$%&'()'*(++%*,%-'*(#.(+%+,/'0123,%4'
13,2'5'.36%7'()'89'$:';'<!"#$%&'()'2(7%/';'
0!"#$%&'()'577'*(++%*,%-'$5*=>&("+-4'? 0!"#$%&'()'*(++%*,%-'$5*=>&("+-'*(+,5*,3+>'13,2'%->%/4!"#$%&'()'577'*(++%*,%-'$5*=>&("+-'0$75*=4@'A!(#$#)!"#$%&'()'*(++%*,%-'$5*=>&("+-'*(+,5*,3+>'13,2'%->%/@'B!"#$#%#& '#$#"#&;<'=#**+ 5%1/!"#"$#%&'($)&&"$#"*($)+,)&"&#-(%&(.(&"%'/0)1-! C%77/! !%*&(/3/! D%//%7!"#$#%!"#$%&"'("$)*+,:3;*4"<=>#8")*A!"#$%"&#$'!"#"$#%&'()*+", %&(-(&"%').*/,-$.*/0"$)*+,
1%2/,&23456"789!
!
!

!"#$"%&'("#)%*+",-&')'("#*-."+/0&./123*4/.#/%*5(6/273*58(9'(#:*-(;/%!%#$#&#CD#8")*A!"#$#%&"$'"()*"+,##./01,2"0%34:3;*4"<=>"?$%2"/,@,/&,5"
A$B,4&",B;,A%"A$B,4&"3.",5+,&"#C")*A

G":1+"7&(@-()*"+,*& !-($%(&6L-(./73(;$#L(MNMNO(PQRQNSTNN!'!"#$%&'!" $%&%'"( ) *"&%' +,)-% .%)/01%2 3)2%$ "* /"("'"-+4)''5+*&)1+)*/ 6%//+ *0,3%12 ."1 (1"-*"2+2 (1%$+4/+"* ". *"*72,)''4%'' '0*- 4)*4%1 ()/+%*/2)%&%*%+,!

78!"#$%&$'(!#

!()*'+)&,-*+.%#/()*%+!"#$%&'()'*+,-%.,/

012

34%'56&/7

89:;<'5#%=-+.>'?;7!"##"$%&'<92@%./(&%=

A1" 9**/3:0/;"3$0<3(/20'0$50#/2"#$%3:02";,/%2/':0&%=2"#$%3:02";," >200=?20"3:$<+/@A.=-.

B%.=%&C+D%<8A

E%#+D%

2F

12%((/34&%2/5%*4&0
61.78-./$&"+0G,+4%H

9F'5<FI2'J7

HH

00'5<1IA'J7

HHH3

20'502I8'J7

HHHK;0'588I;'J7

L.+M+-D+$D%<'51I2'J7!"#$%& #%'()*'+(%(',$-.'/01213$(45-.56&7'89#'1(:6#$';.(<5:<'"$695=#'>81;"?7'9%%@&ABB4-57-$<BCD7EFGEBHFB81;"7IDCJ7KLDMFNO;P!!

01'#(22-&34'/'5-3%/$")()*%+-%.-*/(6'-.'()"#'&H.*",/>'@N'-#+4%'+.='4&(//',"#(&'M(D"#%'
5BNO7
P%:Q"+.,-R-.4',S%'@N'-#+4%'-.,('F'$-,/'4&+6:
D%M%D/
NS&%/S(D=-.4'F'$-,/'@N'-#+4%/'T-,S'M+D"%/')&(#'1',('022'

V+M%D%,:=%U(#*(/-,-(.'+.='=(T.:/+#*D-.4'BNO@+DU"D+,-.4'K%,,- #+*/@(#*",-.4'&+=-(#-U')%+,"&%/')&(#',S%'
T+M%D%,'=%U(#*(/%='
T-,S-.'=(T.:/+#*D%='BNO@(#*",-.4'&+=-(#-U')%+,"&%/')&(#
K%,,- #+*/'T-,S-.'BNO!"7'))* +"/8'#-97:;<8(&'=-.'()"#'& 3(23"2()*%+" BCDEFG/HI=?"(0#/J0"3420( K020/%?3"$<0#/?,/";;*,$<+/LG/'"*'4*"3$%</&03:%#(/?"(0#/%</:$(3%+2"&/"<#/30M3420/J0"3420(/3%/FLN/H033$/&";(" BG/:$(3%+2"&=D/O/+2",/*050*/'%=%''4220<'0/&"32$M/61P-Q8=D/BC/+2",/*050*/24<=*0<+3:/&"32$M/61P!PQ8=D/BC/+2",/*050*/($R0=R%<0/&"32$M/61PSTQ8=D/"<#/G/<0$+:?%2:%%#/+2",=3%<0/#$JJ020<'0/&"32$M/6I1.UQ8=?"(0#/J0"3420()$(3%+2"&

1P-Q

1P!PQ

1PSTQ

I1.UQ

!(

79

>(1'2')-='3%/$%&*)*%+-(+=-.'()"#'-3(23"2()*%+" 9/3%3"*/%J/FBN/J0"3420(/K020/'"*'4*"30#/J2%&/G/K"50*03=#0'%&;%(0#/$&"+0(" -%$J*03B/&%3:02/K"50*03/K"(/";;*$0#/J%2/3:0/K"50*03/#0'%&;%($3$%<
GU+D-.4')".U,-(.W'X

4(.15.(%$26"27"8(0$2'$1"7*(%58*&"*9)9":$&%2)8('/#(&*0+",(-*.*%/0*12'32&*0"$'()*&!"#$%&"$'()***

V+M%D%,')".U,-(.W'Y

!VW

;;

;<

<;

;;

*+

+*

++
!)

!"#$%&'(%)"#*"+*&,-)".)(*$)/#,%'&0$*,#-*!"1*2&"2"&%)"#,3*4,5,&-*."-03*6!789:!"#$%&$'/($+<"3420/K"(/'%<(324'30#/J2%&/J0"3420(/4;/3%/BC!"#$%&$'/('%20(/62"#=('%20(8/K020/;2%#4'0#/4($<+/-@)Q
9**/3:0/J0"3420(-@)Q/K$3:/3:0/($+<"3420

LUNG1!130
LUNG1!182
LUNG1!129
LUNG1!318
LUNG1!061
LUNG1!240
LUNG1!347
LUNG1!003
LUNG1!226%!

=.(&%$1/6*%"8*)5.(8$>*0"12?"382328%$26(.":(>(80"'20*.

LUNG1!198
LUNG1!125
LUNG1!205
LUNG1!211
LUNG1!156

A(%$*6%! # $%&'$( ) " * +," -".
"LUNG1!017
LUNG1!033! " '*% & ' ( %! )&*+,-)"

LUNG1!218
LUNG1!259
LUNG1!232
LUNG1!120
LUNG1!010
LUNG1!055
LUNG1!242
LUNG1!043
LUNG1!292
LUNG1!116
LUNG1!073
LUNG1!107!"#=('%20

LUNG1!258
LUNG1!339
LUNG1!155
LUNG1!026
LUNG1!054
LUNG1!299
LUNG1!071
LUNG1!127
LUNG1!256
LUNG1!268
LUNG1!289
LUNG1!001
LUNG1!300
LUNG1!343
LUNG1!320
LUNG1!119
LUNG1!282
LUNG1!288

GLSZM_SZLGE_LH

GLRLM_RLV_HH

GLSZM_ZSV_HH

GLCM_Energy_HH

GLRLM_RLV_HL!#$%&%'()(&*+(,)-&"!#./)0'1'*/%&%'()(&*+(,)-&!!"# #2(&-*+(,)-&

GLCM_Correlation_LL@*(%58*NGTDM_Strength_LH
GLCM_Contrast_LH
GLSZM_SZHGE_LH
GLRLM_SRHGE_LH
Hist_STD_LH
Hist_Kurtosis_LH
GLCM_Entropy_LL
NGTDM_Strength_LL
GLSZM_LZHGE_HL
GLRLM_LRHGE_HL
GLCM_Contrast_HL
GLCM_Dissimilarity_HL
Hist_RMS_HL
GLRLM_LRLGE_LLGLSZM_SZLGE_LL
NGTDM_Complexity_HH
GLCM_Contrast_HH
GLSZM_SZHGE_HH
GLRLM_SRHGE_HH
NGTDM_Complexity_HL
GLSZM_GLV_LL
NGTDM_Coarseness_HL
GLSZM_ZSV_LL
GLSZM_GLV_LH
GLCM_SumAverage_HL
GLSZM_GLV_HL
GLSZM_ZSN_HL
GLRLM_SRE_HLNGTDM_Contrast_HH
GLCM_Entropy_HH
GLSZM_ZP_HH
GLRLM_RLN_HH
Hist_Mean_HH
Hist_MAD_HH
GLSZM_SZE_LH
GLRLM_RP_LH
Hist_Skew_LL
GLSZM_ZP_LL
GLRLM_RLN_LL
GLCM_Variance_LL
Hist_STD_LL
GLSZM_LZE_LH
GLCM_Correlation_HH
GLSZM_SZLGE_HH
NGTDM_Busyness_HH
GLRLM_SRLGE_HH
GLSZM_GLN_HH
GLRLM_LRE_HH
GLRLM_LRHGE_LL
Hist_RMS_LL
GLRLM_SRHGE_LL
GLRLM_GLN_HL
GLRLM_LGRE_HLGLSZM_LZLGE_HL
GLRLM_LRE_HLGLSZM_GLN_LH
GLRLM_LGRE_LH
GLCM_Energy_LH
GLRLM_GLN_LL
GLCM_Homogeneity_LL
Hist_Skew_HL
GLSZM_LZHGE_LL
Hist_Entropy_LH
Hist_Entropy_LL
GLRLM_RLV_LL
GLRLM_RLV_HL
Hist_Entropy_HL@*(%58*! # $8/
S$+<"3420/6"

GLCM_Correlation_LH

A(%$*6%

LUNG1!206
LUNG1!288
LUNG1!345
LUNG1!022
LUNG1!234
LUNG1!026
LUNG1!342
LUNG1!009
LUNG1!008
LUNG1!024
LUNG1!197
LUNG1!139
LUNG1!153
LUNG1!183
LUNG1!281
LUNG1!131
LUNG1!237
LUNG1!061
LUNG1!037
LUNG1!215
LUNG1!282
LUNG1!295
LUNG1!253
LUNG1!372
LUNG1!178
LUNG1!259
LUNG1!048
LUNG1!233
LUNG1!273
LUNG1!229
LUNG1!060
LUNG1!035
LUNG1!129
LUNG1!145
LUNG1!113
LUNG1!341
LUNG1!337
LUNG1!045
LUNG1!263
LUNG1!271
LUNG1!272
LUNG1!224
LUNG1!046
LUNG1!244
LUNG1!274
LUNG1!294
LUNG1!308
LUNG1!001
LUNG1!177
LUNG1!043
LUNG1!184
LUNG1!142
LUNG1!181
LUNG1!107
LUNG1!258
LUNG1!041
LUNG1!109
LUNG1!005
LUNG1!038
LUNG1!077
LUNG1!117
LUNG1!073
LUNG1!130
LUNG1!179
LUNG1!172
LUNG1!245
LUNG1!140
LUNG1!030
LUNG1!027# $ #3-4*/%&)0%3*305(306--7%&$'#8$9:%" &$'#;%<(30=(*6%>%&7*?@%/3%=A:(0(&*,1&+(B($ &$'#C3%<)0,A=()*&(413%&0>%)0-=*)(&')#D1&+0+%3*)0'(**#E(%)1&(*+(,)-&

2(,"#*/'*)%<*="&.#)%*"9*2')'(5'(+)%*2"9'>)./<*?"%@*ABCDE*FGFA<*HIFFJ*2&#*/'*)%<*2')'(5'(+) 2(#(+)@*HK*FKAAGFKDB<*HIFK!,

coefficient

80
40
0
!40

Sum of absolutecoefficients
Mean error
Sum of absolute coefficients%&&(&'+,
coefficient;2%).)5,%)"#*"+*2,&,.0%0&$*)#*!"1*2&"2"&%)"#,3*4,5,&-*
."-03

" %%

*+,--./0123042,5#6++,+#23 + ,7#-.
7,13 8+,--./0123042,5#&! '* * ) ' "! '*23 + # * 4)'(# ) "&!"#$%&'()*$#+,' ( !%,().$-+&$%)%&/0%&1,,2)!3$-$40+0-)50#+,*!6$(-$'(0)47%+&.%&0+! ( !30'$%+8)+0-4,! 974:0-),*).$+&0'+")&')$)+-$&'&'()2$+$"0+( " !6;)',-4( # !6<)',-4' $ ( !%,( .$-+&$%)%&/0%&1,,2)#$%#7%$+02)*-,4)$)2$+$"0+)=&+1,7+)+10)-+1 .$+&0'+. $% !3$-$40+0-)50#+,- ,.+&4&>02)*-,4)$)2$+$"0+)=&+1,7+)+10)-+1).$+&0'+

40
1 ! 1020
1 ! 101!40
1 ! 100

C&(//EF573-5,3(+'," " # 0#$"" #1 %

1 ! 101

1 ! 103

1 ! 1020
3 ! 10
1 ! 101

1 ! 100
1 ! 100

0.1

0.2

LambdaH5#$-5'-0.3

1 ! 101

Mean error!

G5&5#%,%&

80
1 ! 103! # $%&'$( /) " * +," ""! !"#$%&'()$**'+,%-&,.-$/'.0*.'1,*0&'$/'),/&$2'/3210)*! 455 .-20*')060.-.-$/! 7-8/-#-(,/.'#0,.3)0*'*0%0(.0&'19'2$)0'.:,/';5 .-20*

3 ! 100<)-0&2,/'=>'0.',%?'='7.,.'7$#.@?'A545BCCD'4EAA
1 ! 10080

0.1

Lambda

0.2

0.3

!$

?@"()*%+-%.-$(#)*(2-2*A'2*4%%=!"#$%&'(&)*+&,$-.%)*&$/&!()("0("1)*&!$/(2).'+&3$*4&56789&:;:5+&<=::!"#$%&'!%?1(2"()*%+-/')4%=)66(5&3$3(4:(U(MNV8(

X*&":":C(5&3$3(4:(U(TJV8((((Y&6"#&%"1:(5&3$3(4:(U(ZN8

E1:3%*,5%"1:(12(E1F(>*1>1*%"1:&6(/&W&*#(+1#$6(
4E.!;8(K"%/(%/$(3"C:&%,*$
E&65,6&%"1:(12(*&#"1+"5(351*$3(4*&#S351*$38(
D&3$#(1:(51$22"5"$:%3(12(&(E.!;

E&65,6&%"1:(12(*&#S351*$3(D&3$#(1:(51$22"5"$:%3(12(&(E.!;(2*1+(&(%*&":":C

E&65,6&%"1:(12(+$#"&:(12(*&#S351*$3.&%"$:%(3%*&%"2"5&%"1:(,3":C(+$#"&:3(D&3$#(1:(%*&":":C(#&%&3$%.*1#,5%"1:(12(@&>6&:S;$"$*(5,*9$3(&:#(5&65,6&%"1:(12(>S9&6,$3(461CS*&:'(%$3%8!&

B'&"2)&5-3%+&)*)"'+)-.'()"#'&-*+-)4'-&*6+()"#'&
K%,,- ."#$%&'5K!7
BXG[C\G[XB]\?0
BXG[C\G[]\0F
BXG[C\G[]\<1F
BXPXC\GPYB]\8
BXG[C\G[YB]\;2
BXPXC\GPYB]\;8
BXPXC\GPYB]\<<<BXG[C\G[YB]\<11
BXG[C\G[YB]\<10
BXG[C\X[YB]\;2

V+M%D%,:=%U(#*(/-,-(.'5VZ7BXPXC\XP]\YX
Y-/,\C%+.\YX
BXPXC\XPYB]\XX
BXG[C\[^\XX
BXG[C\X[]\XX
BXG[C\X[XB]\XX

FGD2:# F&%H 3(+(3 <0>( >-=( '%)&0I
FGJG:# F&%H 3(+(3 &1= 3(=4)6 '%)&0I
D2GFC# D'%33 >-=( 3-K 4&%H 3(+(3 ('/6%<0<
D2C# D'%33 >-=( ('/6%<0<
DJ9FC# D6-&) &1= 6046 4&%H 3(+(3 ('/6%<0<
D29FC# D'%33 >-=( 6046 4&%H 3(+(3 ('/6%<0<
G29FC# G%&4( >-=( 6046 4&%H 3(+(3 ('/6%<0<90<)# 90<)-4&%'
GJC# G-=4 &1= ('/6%<0<
GJ9FC# G-=4 &1= 6046 4&%H 3(+(3 ('/6%<0<
2$# 2-=( /(&,(=)%4(
G2C# G%&4( >-=( ('/6%<0<
G2GFC# G%&4( >-=( 3-K 4&%H 3(+(3 ('/6%<0<81"'

B'&"2)&5-#(=<&3%#'&-(+=-&"#1*1(2-)*/'
V+M%D%,:$+/%=')%+,"&%/'5VE7'

K%,,- ."#$%&:$+/%=')%+,"&%/'5K!7'4

4
4-./012345456

surv@"AB;."0C

Rad!score
Rad!score

Rad!score

D$2;."0C

2
1

6
5
4

>'?")%

3
2
1
0

!1

6
5
5
4
4
3
3
2
2
1
1
0
0

1
1

!1
!1
4

7)'(/#'-./01781

7)'(/#'
9:/01781

3

3
surv

Rad!score

Rad!score

surv
surv 6

2
2

0
0

0

4

9:/012345456

333!"#$%$&'()%*+(,-#./

903.-8,+6
!+5
$"

2

6
5

surv

2

2
1

6
5
4

4
3

1

3
2

1

1
0

0

0

0
!1

!1

7)'(/#'"#

7)'(/#'

B'&"2)&5-$<1(2"'&-%.-C($2(+<D'*'#-3"#1'&
V+M%D%,:$+/%=')%+,"&%/'5VE/7'
5@(.M%.,-(.+D')%+,"&%/7

K%,,- ."#$%&:$+/%=')%+,"&%/'5K!7'
++
++

0.75
0.50

++

0.00

M&%0=0=4

9046A&0<5N%307%)0-=

+++
++
+++++
++ +++++++++++++++++ + ++

0.25

++

++ + +

01

1.00

G-KA&0<5

2

+++
+++++
++ +
++

3

4

D1&+0+%3*)0'(*?H&<B

+
5

+

+

D1&+0+%3*/&-L%L030)H

D1&+0+%3*/&-L%L030)H

1.00

0.50
0.25
0.00

6

+++
+
++
+ ++
+ ++++++++ +++++ ++ ++++++++ +++++++++ ++ ++++ + +++++ ++
+
+ +

0.75

0

1

2

3

4

D1&+0+%3*)0'(*?H&<B

5

6*:M+D"%
5S+R+&='&+,-(_';2J'U(.)-=%.U%'-.,%&M+D7
N&+-.-.4'5.`<927

O+D-=+,-(.'5.`A17"#?IA'a <1;, 51I8<_'1I0?b1I?27

9I8'a <1;) 51I90_'1I<?b1I?07

VE

2I;'a <1;" 51I2A_'1I9Ab1IFF7?IA'a <1;# 51IFF_'1I8Fb<I?17"!

<03,2$0 2&0-)(%,=)3)%> "+ %"2"3"/)(,3 $)/#,%'&0 "# 2&0%&0,%.0#%
23,##)#/ !?*).,/0$ "+ $%,/0 @*#"#A$.,33 (033 3'#/*(,#(0&*2,%)0#%$=0+"&0 %&0,%.0#% B)%4 $%0&0"%,(%)( ,=3,%)C0 &,-)"%40&,2>:;,30<0#=>#?+2<@+0 A>#64#01B#!"#$%& '%(&)$B#'(''C=0D@<2#;,30<0T>#A23640D0#?+2<@+0M>#E@D,#FG2+0D0H0J;6540#I25,<2J0H>#=030<0-0 E,-G240D6V>#E,-G2J@D2#FG2,J0<0P!

!"#$%&'"(&)*+),"$-&.)/01"(0"23)4%$56$&")/0.**-)*+)7"510$-)/01"(0"23)8962.6):(1;"%21&9"!1;121*()*+)7"510$-)<6$(&6')/01"(0"3)!"#$%&'"(&)*+),"$-&.)/01"(0"23)=$06-&9)*+)7"510$-)/01"(0"23)8962.6):(1;"%21&9#>$&1*($-),*2#1&$-)?%@$(1A$&1*()8962.6)B$(0"%)B"(&"%$/$(+*%5)C6%(.$')D%"E92 7"510$-)!120*;"%9)F(2&1&6&"%!"#$%&'"(&)*+)B-1(10$-)G$51*-*@93)4%$56$&")/0.**-)*+)7"510$-)/01"(0"23)8962.6):(1;"%21&9&F*()C"$')H."%$#9)B"(&"%3)/I4I),F7IH)=*6(5$&1*(""

827.)012'3%*! !"#$%#&'()'*'+(#),',#%*,-%.,'/0,(/.'+/#'0*,(%.,)'1(,2'),*$%'3'./.4)-*55'6%55'5".$'6*.6%#'78!9:9;<'*.='),%#%/,*6,(6'*>5*,(?%'#*=(/,2%#*0&'7!@AB;'()'#%6/--%.=%='+/#'(./0%#*>5%'0*,(%.,)4! C",6/-%)'/+')"#$%#&'*.='!@AB'+/#'),*$%'3'8!9:9'0*,(%.,)'
1%#%'6/-0*#*>5%A>C! D#/>*>(5(,(%)'/+'5/6/#%$(/.*5'#%6"##%.6%'7:BB;'*+,%#')"#$%#&'*.='!@AB'1%#%'*5)/'6/-0*#*>5%A>C
D#%=(6,(/.'/+',2%'6*.6%#'#%5*0)%'>%+/#%',#%*,-%.,'()'(-0/#,*.,',/')%5%6,'*'-/#%'*00#/0#(*,%',2%#*0&
JK L$0M("%)GDK)!"##"$!%&'()*+"),*+#%-.' #)(/0'+"0,1K)NOJPQ)JRSTUV)ROTWRXRK)NK I0M"%2*()C4K)2&(3+#)(/0'K)NOJYQ)JNRV)JYRWJZJK)XK ?(12.1 ,K)4(,+!+5)67), 8(/%*+97%*+:;<=K)NOJJQ)YJSRUV)JXRNWYK"(

E+1'#)'=-7'))* +"/8'#-/($&!"#$#%&'()#%&"*(#+&$,

DN(+&>

1,//# %2+),"!"#$#%&'() *&+)3(4(6)6(4(7

DT(+&>

23+3'3$#4&''510#66.".%/17+&4.7!!

"DN(+&>,%-."/.0()1*&+

1,//# %2+),"#)3(4(5#)6(4(3-%.,"/,0)#%&"*#+&$,"DT(+&>!!" #$%&'()*+"*,-./0'1$*2/.3()-.%4*5)(--6*7889"*")

832,'-".:$(/(34#,/"$&0#/3%/0M;*%20/3:0/;20#$'3"?$*$3,/%J/3%;%*%+$'"*/($+<"3420(/*$<X0#/3%/3:0/*%'%20+$%<"*/20*";(0/6P!!8/"<#/#$(3"<3/&03"(3"($(/6UQ8/%</;20320"3&0<3/;*"<<$<+/'%&;430#/3%&%+2";:,/$&"+0(/%J/(3"+0/W/<%<=(&"**/'0**/*4<+/'"<'02/6IS-P-8/;"3$0<3(/?0J%20/320"3&0<3/K$3:/(3020%3"'3$'/"?*"3$50/2"#$%3:02";,/6S9H!8Y",83

96$%$).6($%#'2+.&$'%!"#$$%$&'()*+,-./'-)*)&0#+12'%*#&.3')4'567'
3-#&.'8'9:;<;'+#-%.$-3'-0.#-./'=%-1':>?@'#-'
A2,31,'B$%C.03%-2'D)3+%-#"'=.0.'.*+")2./

6&-'(7-#"8'9$%'2 9#:'(9-;$#%,,
B-%;-"
C#4I-9#4K/#&L6
LM
N$./O4O&7
6;-%OF#"F$%O9#
KH5#9O5.'F-44'F#"F$%O9#
P#"&-'F-44'F#"F$%O9#
QO91O%-%/
KO4$;'
R#"/'.O4$;
BBS!59O"'3O459-'(99!8'9$%'2 9#:'(9-;$#%,,
R"O&%O.$.
PTT
PTT'U"-!!PTT'(9O%/G8'9$%'2 9#:'(9-;$#%,,!"#$%$%&'(%')'*+,*0'2 <='(>?,!-./'(%')'*0,*0'2 ?<'(><,

123#450@+?*'(A'/-./,

DD
J=

D*
=D

0@E+<'(FG$2.H5#"-,

DE
JJ

ED
J*

0@E*+'(FG$2.H5#"-,

D=
J=
E

E*
JE
=

0@ED?'(FG$2.H5#"-,

+J
>
*

DD
=J
D

0@EE+'(FG$2.H5#"-,=D?D'2 +>E>D'(??D=, =0=+'2 D+DDD'(<J0=,

0@*<?'(A'/-./,

=?
D>

=*
DD

=@00'(FG$2.H5#"-,

E@*'2 =*J@J'(EJ@J,+@E'2 =+?@J'(J<@?,

0@??*'(A'/-./,"$7'))* /($&-.%#-/"2)*$2'-)4#'&4%2=*+6-$*F'2-1(2"'&7,#&#8%/ =>?$%@?
7,#&#8%/ 9#8%,: #$%&' 3;4+5!" #$%&'

6
(

6

6

"+,'-+./0
1%/2' 34+5

)**

9( 6

6

6

9< 6

6

6

9<( 6

6

6

(

4+

)**

A81',4'0 =>?$%@
B%1'/'4?0'C.$@.-#4#.8
A81',4'0?9#8%,:?#$%&' 3;4+5
!!!

""!

"!!

"!"

!"!

!""

!!"

6

6

(

"+,'-+./0
1%/2' 34+5

"""

6
)**

=> D?='44# 82$9',

#9( 6

6

6

#9< 6

6

6

#9<( 6

6

6

(

4+

)**

"%

!"#$%&$'()#*+,)'-#.-+.#*%/&
5)6*&+)02-2#/7.7()*)+",)-).*/0)-12-3)4)-*
L.(:(#)%*(,):/

M)$/%/'*(,):/

N/''( #&,O/.*,)-.

-/

-/0-.

!"#$#%&'!"#$%&'() *+,-./012*3-./4/12*+5-(%)*"+*,./6712*+5-

8.9:12*;-!"#$%&$'()#*+,)-)*),.$)&)%*/')&0#&1)")&*

2.3'.('()#*+,)4564

5#7)8)*'9)1.":.4%*%.&

;<=

>64

?,%$%&#8'>2'"#::%&$;@AB=PQTQR*3*P.)SG%/$/%*.&#G%/#:'8*,)'.(;%>64?,%$%&#8'C'%&7),*)9'>2'"#::%&$<BDD<

PQ2UR*3*P.)SG%/$/%*5(6/G6"#/*,)'.(;>564

E."F%&#*%.&'G564'H'>64I

;@DA<

%>564

E."F%&#*%.&'G564'H'%>64I

<J;JK

84

PQ!R*3*P.)SG%/$/%*+"G"++&../#+/*,)'.(;VPWXR*3 V/(:8O".*:.)SG'"#/*0(99/./#+/*,)'.(;"&

8+''% (9/*+2):8;<)&(3)%(=+2'+3)8;! C%77/! !%*&(/3/D")C-),.")+@(?B('#"/-%%.)A(.)(9.)+#3(.*+A-! D%//%7

B-=(?B('#"/-%%.)A(.)(.)C-#,-1(9.)+#3(.*+A-('@1#&+&(X-()*"+,*&(!-($%(&6L(X/1*&5(E&:5$*L(MNMM(),COTJ4TV8RMTTZSMTMP

5&)$"1),(6$)4(*"'&/,"(&1#(*"'&/,"(7*""('$18"#().(,$+1$7$%&1)(7"&)9*",(.7($1:"*)"#(3;(0&/(7"&)9*"(<$3=>(7.*(?@@(&1#(A!

!""#$!%&%'()*%+,-$'(-,./(
01#$0*/2,+2$3(2,/2,/*/!"#$%$&'(&)*+%,*$&-(&./&$01&'2"*$3&4$53.*1&6766&),89:;<:=>?6::@A6:6B(#

7&0-)(%,=)3)%>*"+*D*$)/#,%'&0$*+"&*3"("&0/)"#,3*&03,2$0*6E<<:*,#-*-)$%,#% .0%,$%,$)$*6F9:*-./.0)1".&%2'0)2%3#)'*-44,'!"#$%&$ ()$%#$%#"#'*!+,'!"#$!"%&'&()!"#$!"%&'&()(!

85

!"#$%#&'()*+#,-"*#.*(%/-"'-0*123*45+65%78-(-"*)9"/-&*#.*'(:-*'#*
6#)#"-$(#%56*"-65+&-*;<==>*5%0*0(&'5%'*:-'5&'5&(&*;?8>!+C-@-,(0-+,2#-%?-,,. )2*9-#(0-+,2#-%
E(=+C-@-,(0-+,2#-%

F)C-#,-1(?-,,. )2*9-#(0-+,2#-%
E(=+C-@-,(0-+,2#-%("@1#&+&(X-()*"+,*&(!-($%(&6L(X/1*&5(E&:5$*L(MNMM(),COTJ4TV8RMTTZSMTMP

G"//(#H-9I;!8%/%1"-%*%"02#+*/%*2+/+%2#9+/:+*#4)*4%1#4"0'$#4",%#.1",#4)*4%1#(1"(%1/+%2#;4%''#$%*2+/5<#*%41"2+2<#)*-+"-%*%2+2<#-')*$#$04/2<#%/4=#!8"'%#)*)'52+2#;/"("'"-+4)'#1)$+",+42=#4)*#3%#)*#%>(')+*)3'%#?@

5&6'78)#$".&'.0'/.(()&$#'9((

86

!"#$%&"'(")(*+,&-*+,./%(0"#(.)12%,#345647(8-',-*9-#(4:4;(<!+#%+=>?+%.%("0(*+,&-*+,./%(.)()+)"*-1./.)-(%,#2/,2#-%(+)1(@.0-(%-)%.)A!"#$%&'&()*"+,*&-(./0.*12$331*((4+$#"5&6(./73"538
0"9"3"1:(12(;$#"5&6(<,&:%,+(=5"$:5$0$>&*%+$:%(12((!$&6%/(=5"$:5$3-(?&5,6%7(12(;$#"5&6(=5"$:5$3@7,3/,(A:"9$*3"%7!"#$%&&'()*&+,#!"#$%&'()*+,-./%'./%+'/0,10("%%%2./$%34/0('5%6'()*+,-./%7,+%#,8,5,*0('5%+'/0,10("! 9+/$%:8850('#0,."%,7%#,8,5,*0('5%+'/0,10("!

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
6<$"("*-"V=,<$0"%-'E"%+&(".'4/+%%+".'K"/",()12=W:"/1$&-*'K"/",()12='T/),-%'"3'D-)*').*'8-0U':).0-('E)&+-.&%X'@;-'Y:B'C(+/$() DB'-&')<R'Q-&)#"<+&-%'GHGGJ! 9$//)(='@SJ"

87

-(.)/*'$%0+(1#,(2+3/ %44$(%35'*('+0#,*+6+3%*+(,'(6'789-'2)*%*+(,/'%2(,1 4%*+#,*/':+*5';<=>='
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
D"#E%4&3('),(;#9%7323+%5#:;8408#<+3=%(43);#.,4&3)'/
F"#$%&'()*%+)#,-#9%7323+%5#?'28/);#,-#9%7323+%5#<+3=%(43);#,-#9'/';'
G"#$%&'()*%+)#,-#9%7323+%5#?'28/);#,-#9%7323+%#'+7#.%'/)0#123%+2%5#<+3=%(43);#9'/';43'#1'('H'I

B":1+"7&(@-()*"+,*& !-(.CD=(DBE(FGFH!"#$%&'()*+(,-(%'*.(%/0%1234%56.*."/,7%",%89:;:%*5/,#%<7"*,!"#$%

456!Z'-1+*-(/)<',("7&2'3)0&"('(-0-1&"('
89:;:Z'.".L%/)<<'0-<<'<$.,'
0).0-(&$'($"#$%
$#+$%
&'()

"<$#+%=)-(!"#$$%
&'()

8#/%
;&)8

+#8%
)&:1
234567
8#/%
)9.+
/#$%
234567-0,12345679#(:$%;#;)<%-)=>?>@?#%$)

9>?@A4
9>?@A4%<)4ABC)#;)D>-<%-&>=)

8"#/"%,)-.

E')FG>?H)E#%D;'+)#I>I+)$>>=J>)E#%D;')!?#;;.>)

*#+"%
,)-.

!""#$%&'#()*%+#,%#-*.&%/*
01(#$2(,+()*%,+3#%#-*-+2345*%)+14&6#*
7&6,4# &6/&8&%1+4*

8#/%
=)-(

8#/%
;&)8

+#8%
)&:1
234567/#B%-0,1234567?>;?8)%-)J#K.#=)E#%D;')!EJ%%=)?>;?8

+#8%
)9.+
234567)#*(+,(-./(0123,(456,,("7(8!9#7:(#"%7;#"&<,&

A$#3+/+(,'2#0+3+,#! 0->@#;#%$)&>=#@#$>L)()<%-&)%<)&>=#@#$>)?M(?).;>;)#$<%-&(?#%$)(E%.?)()D>-;%$N;)%G$):>$>;)%-)D-%?>#$;)?%)D->O>$?+)=#(:$%;>+)%-)?->(?)=#;>(;>;)! 0(?#>$?;)G#?M)?.&%-;)?M(?);M(->)();(&>):>$>)&.?(?#%$)@($)->@>#O>)();(&>)=-.:)?M(?)?(-:>?;)?M(?)&.?(?#%$+)$%)&(??>-)?M>)?'D>)%<)@($@>!M??D;LPPGGGI@($@>-I:%OPD.EJ#@(?#%$;P=#@?#%$(-#>;P@($@>-Q?>-&;P=><PD->@#;#%$Q&>=#@#$>8'88

=776(7%/,%>?(.>%@"/&7A%0/'%-*,-('%.'(*.5(,.%%!"#$%&#'("'')*'(+&,-%.

" 9"/-'1)&+-.&%'/)='(-3$%-'&2-'+.>)%+>-'.--*<-'#+"1%='*$-'&"'0".0-(.%')#"$&'1.-$/"&2"()V'[(-3$%+.,'()&-Z')("$.*'SHL\H]'@6$U$+'KB'K2"()0+0':).0-('GHIN^IHZPHI_`Ja" ;+b$+*'#+"1%='@0+(0$<)&+.,'&$/"$('M8CZ'0&M8CJ'2)%'."&'%2"7.'2+,2'%-.%+&+>+&+-%'[%-.%+&+>+&=Z')("$.*'PH]'@c02+*)'dB':<+.':2-/'
GHIP^eI@NJZIINI_IINe Ja" C'%+.,<-'#+"1%='"3'2-&-(",-.-"$%'&$/"(%'0"$<*'<-)*'&"'$.*-(L "('">-(L-%&+/)&+".'"3'"/+0%'+.3"(/)&+".'[5-(<+.,-( QB -&')<R'8'4.,< d'Q-*'
GHIG^SeeZOOSLNGa" 8"(/)<'&+%%$-'%2"$<*'."&'#-'%)/1<-*'/&01&)(+&,-%.

7!"#$%&$'(%)*+,')-'./-,.'(%)*+,''" 8".L+.>)%+>-'>+(&$)<'#+"1%='&2)&'-V&()0&%'72)&')(-'-b$+>)<-.&'&"'+.3"(/)&+".'"#&)+.-*'3("/'!"#$%#&'(%&)*%+',%#,*-%".(/*-'()0*"#*1'0%"+%2(!"#$%&#'("'')*'(+&,-%.(+.(-$/0,*,1&%/%0".>-.&+".)<'+.>)%+>-'''''f7-&f'#+"1%+-%B'3("/'/-*+0)<'+/),-%'3"('0).0-('&(-)&/-.&'" :"/1$&)&+".)<'1("0-%%-%'&"'02)()0&-(+g-'&$/"(% )%'7-<<')%'!"#"$%&'()*%+,."(/)<'&+%%$-%'#='-V&()0&+.,'+.&(+.%+0'!"#$"%&'"()*++.3"(/)&+".'3("/'/-*+0)<'+/),-%<

01#2'/)'!'#++3"4'%&'-#/%)"%5+6+/"$,-01"'
234$3"&',$'56'.74%"8! 3#$@>)?M>)&>=#@(J)#&(:>;)@%.J=)E>)?>-&#$(J;)%<)@(;@(=>;)<-%&)%.-):>$>;+)?M>)#&(:>;)&(')#$@J.=>):>$>?#@)($=)D-%?>#$)#$<%-&(?#%$I!"-4<,=.-"! A>$>?#@)($=)D-%?>#$)#$<%-&(?#%$)(->)@%$;#=>->=)(;)#$@J.=#$:)D-%:$%;?#@)#$<%-&(?#%$+)($=)?M.;)&>=#@(J)#&(:>;)&(')(J;%)#$@J.=>)?M>&I+&,-".$!"##"$%"&'()*+/"$,-01"'
234$3"&',$'+>6'.74%"8

9,73-3%7#-(,*#0))&"JJ,*324KL&J'M,8)N,*324
9,73-3%7#-(,*#0))&"JJ,(C'+32'=3()8'/%K'/)%(=34)'K,(CJHHH>K20%*34)(;K*48K%78J-'28/);J(%8420JO3()PQ)R*/J+82'2374K0)*

9"$"':.-/'
7;-4-.,$?(,+)#B**8+,/K#>S!G#?%M#>TUV"DDK#7,3"#!SK@@WTJ-3**8K>S!GKSSSDDK#

89=

;%,(B ($'2+3$(B($0#$'+,6($2%*+(,'2+15*'.#'(.*%+,#0'6$(2'2#0+3%&'+2%1#/'C'!"=5%$%3*#$+/*+3/'(6'789-'2)*%*#0';<=>=! !"#$#%&'()&**+*,"(-"#*./&",%.#01('.//&"&0%.,%.#01(,0'($.2",%.#0(#/(0#034$,**()&**(*+02(),0)&"(567898:! ;0%",3%+$#"(<&%&"#2&0&.%=(#0()#0%",4%(&0<,0)&'(58>:(8?(.$,2&4!
! !"&4&0)&(#/(2"#+0'32*,44(#-,).%=(5@@A:(),+4.02(*#B(.0%&04.%=(),C.%,%.#0!"#!"#$%&$'()*$+,-(./.01&$234$524"67$28"9234&$!2#$:&$;<,=0*$>=?&$@&$A/(=&$B=C./&$2D$524"E7$6DE96FE&$!G#$@&$H,(*$')=0$A,=CIJ&$E3$524"67$22923&##!"#$%&&"'()*+,#&

90

78(%2'%"#$4+'$4&4-#24/'9-)"':;'%"#$4+

IJK"%("+&L$3(L$:$*&%$# 2*1+(MN("+&L$3(,3":L(21,*(*&:L$3(12(!1,:32"$6#(,:"%3(4!A8(12(MN("+&L$3(4J
HGGG(%1(HOGG(!A-(JHPOG(%1(HOG(!A(
Q6,:L(*&:L$(R":#1RS-(&:#(JHOG(%1(
FOG(!A(Q+$#"&3%":&6(*&:L$(R":#1RS8(
R"%/(21,*(K"%J#$>%/3(&2%$*(*$J
I,&:%"T&%"1:(4O-(U-(V-(&:#(W(K"%38X#'

4(&'(7(,.*.")(%"5*#(7%/0%:B%"5*#(7C%@",*'A%"5*#(7C%*,D%E(..",65@('%5*&7
Y$>*$3$:%&%"9$("+&L$3(12(4&8 51+>,%$#(%1+1L*&>/7(4MN8("+&L$3-(
4K8 K":&*7("+&L$3-(&:#(
458 Z$%%" :,+K$*(4ZB8(+&>3#7

4(-(")('%/&('*.",#%-$*'*-.('"7."-%F4G:H%-6')(7%0/'%"D(,."0"-*."/,%/0%(&"D('5*+%#'/?.$%0*-./'%'(-(&./'%56.*,.7
Y$5$"9$*(1>$*&%":L(
5/&*&5%$*"3%"5(4YDM8(5,*9$3(21*("#$:%"2"5&%"1:(12($>"#$*+&6(
L*1R%/(2&5%1*(*$5$>%1*(+,%&:%3(,3":L(Z$%%"(:,+K$*(
4ZB8J-(1*"L":&6("+&L$(4D[8J&:#(R&9$6$%(#$51+>13"%"1:(
4\08JK&3$#(+1#$63(R"%/(&*$&(,:#$*(%/$(YDM(5,*9$3(4)AM8(":(4&8(%/$(9&6"#&%"1:(&:#(4K8(%/$(%$3%(>*15$#,*$3#<91

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
D"#E%4&3('),(;#9%7323+%5#:;8408#<+3=%(43);#.,4&3)'/
F"#$%&'()*%+)#,-#9%7323+%5#?'28/);#,-#9%7323+%5#<+3=%(43);#,-#9'/';'
G"#$%&'()*%+)#,-#9%7323+%5#?'28/);#,-#9%7323+%#'+7#.%'/)0#123%+2%5#<+3=%(43);#9'/';43'#1'('H'I<4=#2%)&+1%*'(42>44&'"4/%5#='4?%/4&54+'#&/'2)*)=)$,! 2.?D.?;)%<)?M>),V);M%.J=)E>)>WDJ(#$(EJ>! V$D.?;)?%),V);M%.J=)E>)E(;>=)%$)&>=#@(J)>O#=>$@>;
9#-&$,(*%,+3#%4*=&1(13&$,(*$10;16#6%4*&6*/&4%1(13&$,(*&0,3#4:1;1(13&$,(*$10;16#6%4*&6*<:*&0,3#4" D+$,0(E#'=" 8&**4" ;4*,0'4" 8,0)&"(" 6&)"#4.4" D#*&4(" @*,0'('+)%(

5%+E&4:" 8,C.%.&4" F&44&*!!

JE'*(4(&(1+3%&'%,%&?/+/'!"

92;)*)=)$,8(#+4/'-#/%)"%5'94#23-4+'9)-'*-4/%52%)&')9'*#-)2%/'$=#&/'5#&54-'"#=%$&#&5,'$-#/4'%&'"#$&42%5'-4+)&#&54'%"#$4+@TU$%2+/) AB'C(+/$() DB'-&')<R'Q),.-&+0'!-%".).0-'Q)&-(+)<%'+.'E2=%+0%B'?+"<",=').*'Q-*+0+.-'GHGSJ*%X#-% VY.;M#&(H-FZ/#=>?(Y(),-#&.-(PZC'.X#)[(;.&(?;.]Z/#=>&# *(&>\(G(OZ*>$?()"#$%&#'(U!"#$%&'()$*("+,"-$&.(/"012$*1$34"5'&67&($"01/++."+,"8$621&."012$*1$34"9:73/7";*2<$'32(:4"=>!>!4"8&26&3/24"-2?&3/2>@74"A7@7+@&"B!C>BDBC4"E&%&*
C"#$%&'()$*("+,"F&62+.+?21&."G$1/*+.+?:4"H&)&?71/2";*2<$'32(:"-+3%2(&.4"!>!>!"82*&)2>@+?73/24";I$4"H&)&?71/2"JDD>BDKD4"E&%&*="#2<232+*"+,"L7&*(7)"F&62&(2+*"012$*1$4"#$%&'()$*("+,"-$&.(/"012$*1$34"A&17.(:"+,"8$621&."012$*1$34"9:73/7";*2<$'32(:4"=>!>!4"8&26&3/24"-2?&3/2>@74"
A7@7+@&"B!C>BDBC4"E&%&*
M"#$%&'()$*("+,"N(+'/2*+.&':*?+.+?:>-$&6"&*6"O$1@"07'?$':4"A&17.(:"+,"8$6212*$4"92*6&2 ;*2<$'32(:4"=JJ>C4"N*+/2?&3/24"0&:&)&4"N3&@&"DBP>KK!M4"E&%&*
D"#$%&'()$*("+,"F&62+.+?21&."G$1/*+.+?:4"A&17.(:"+,"A7@7+@&"8$621&."G$1/*+.+?:4"G$2@:+ ;*2<$'32(:4"Q>CC"823&@2>)&1/24"N)7(&4"A7@7+@&"B=Q>BDKD4"E&%&*
Q"0&*,+'6"R7'*/&)"S'$I:3 8$621&."#231+<$':"T*3(2(7($4"!KPK!4"O+'(/"G+''$:"S2*$3"F+&64"U&"E+..&4"VW"PCK=J4";0W!'-%#+)*.$/0%,.$&%0*1%#2! K2-'&2(--'/)h"('%)<+>)(=',<).*%')(-'&2-'1)("&+*',<).*B'%$#/).*+#$<)(',<).*B').*'%$#<+.,$)<',<).*%R! 9)<+>)(=',<).*%'1("*$0-'%)<+>)B').*'&2-'2*$")()134%(541""'*%6(3$.0&+".')%'0".*$+&%'3"('*-<+>-(+.,'&2-'%)<+>)'&"'&2-'"()<'0)>+&=R!
! E)("&+*',<).*'0).0-('@E5:J'+%')'()(-'3"(/'"3'0).0-(B')00"$.&+.,'3"(')11("V+/)&-<='P]'"3')<<'2-)*').*'.-0U'0).0-(%R5!"#$%&'(#)*#+,#&-"#$.,&#/01,(.2+% 34,(.2+% 56!55"#32&'7#89*#+,#&-"#),(-&:4'7(- /+&;#<+.=#>?:7#56!@
A"#/(#B* +,#&-"#/+&;#C#<+.=#)'.(-(74#56!!

!"-%#+)*.$/0%,.$3%,34#$5-678$! K2-'">-()<<'%$(>+>)<'7)%'%+,.+3+0).&<='7"(%-'+.'
1)&+-.&%'7+&2'2+,2',()*-'0).0-('&2).'+.'
1)&+-.&%'7+&2'<"7'&"'+.&-(/-*+)&-',()*-'
0).0-(!R! K2-'&(-)&/-.&')11(")02-%'3"('1)("&+*'0).0-('*-1-.*'".'&2-'/)<+,.).0=',()*-'"3'E5:5R'! K2-'/)<+,.).0=',()*-'+%'*-&-(/+.-*'#='$%+.,').'+.>)%+>-'3+.-L.--*<-')%1+()&+".'0=&"<",='@68C:JR! K2-(-3"(-B'b$).&+&)&+>-').*'.".+.>)%+>-')11(")02-%'%$02')%'()*+"/+0%')(-'1(-3-()#<-'
3"(')%%-%%+.,'E5:'/)<+,.).0=R".,+7'#: %)#'/K#!"#$#%&'()*+,-'(.#>S!W5#@TX!Y5#GFNVS

Z3403I'7,#[#%)#'/K#/,&#$#0(1,#%,2'(5#>@XDY5#G!FNG>D5#>S!WK#

93

!14#%00$&394'4@422% "#*+'9-)"';A'#&/';B'>4%$124/'C<'%"#$4+'9)-'D'*#2%4&2+'>%21'1%$18 #&/'=)>E%&24-"4/%#248$-#/4'FG:!"#$$%&'()*$+'*,'$-%'.*+$',%#+/0)%',%#$1&%+'*0$#/2%3',&*.'$*(*)*4/"#)'#23'"*25%2$/*2#)',%#$1&%+94

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
8+4/0(#0,.%,7%8'#04.#"%A0#<%@B%('.(4+%6=%-"0.*%?C;DE;%01'*4"F%;<0"%0"%#<4%70+"#%"#-/=%#,%4G'10.4%#<4%8,#4.#0'5%,7%?@%74'#-+4"%,.%?C;DE;%01'*4"%7,+%8+,*.,"#0(%8+4/0(#0,.%,7%8'#04.#"%A0#<%@B%('.(4+F%!"#$%&'"%

Z"*%/(12(&(/16$

N/*$3/16#(
9&6,$(^(+

K2(-%2"<*'>)<$)&'*-)&2-4#&*&)4,)$9+'+0+/2$:*)9$3+,&*.4#*,/$&*;4$

E-(%+%&-.&'*+),()/+

0$&%/(12(&(/16$$

N/*$3/16#(
9&6,$(^(9K2(-%2"<*'>)<$)&'#+(&2>/,%*?@*$,6*$,;%2+#A*?%D%J%:#@(JJ')#$?-#$;#@)D-%D>-?#>;)(;;%@#(?>=)G#?M)?.&%- M>?>-%:>$>#?')G#?M)->;D>@?)?%)$.&E>-)%<)@%$$>@?>=)@%&D%$>$?;)($=)M%J>;+)($=);#\>

95!#

-<$*'%/4&$=#+'$-<$.*%/#%'&!"#$! "! # $! %"$&$'($)*&+,&-$*.$#/"$%*,#/0)"&#/$122,)*.&#"3$ "! # $! # %& '# (# ) # * 4/","$* )".2#"3$#/"$.5-%",$26$7&*,38$! "! # $!
4&3$-&77")$*.#2$! "! # +! , $! - "! *.$#/"$%*,#/07",3*3#".1"$122,)*.&#"38$9":#;$! "! # +! 4&3$#,&.362,-")$*.#2$'($*-&+"3;$/" 01# 238%</"$=&533*&.$)*3#,*%5#*2.$1&.$%"$":7,"33")$&3>/" 01# 23 , 4 5& +! 6" '! ()! 1# 2!#$'
1 - "! * < 2 - +! *6" '! ()! 1# 2 ,9:; =! "! # +!
(8 *
(78 *
4/","$8 )".2#"3$#/"$?@;$&.)$1 &.)$2 &,"$#/"$,24$&.)$12A5-.$26$&$7*:"A$2.$#/"$'($*-&+";$,"37"1#*B"AC8D$A*."&,$4"*+/#*.+$65.1#*2.$#/&#$1&.$&)E53#$#/"$*-72,#&.1"$26$7&*,3$*.$)*66",".#$,"+*2.3$1&.$%"$":7,"33")$&3>>#+! ? >+!
#> A +! A @5& +! ,@'#+! B @
4/","$! *3$#/"$)"7#/$26$#/"$F5&.#*G")$H<$2,$'I<$*-&+"3$J"8+8;$'$K$LMM$*6$#/"$*-&+"3$4","$NO%*#$)"7#/P8!"#$%&'"'

I('7"7.(,.%$/5/+/#A%"5*#(7! &#''/$0/.%$#9D#'/'0C>$4/*#"$,"+*2.3! ,#B/C>$%A&1Q$,"+*2.3!"#$%&%'#()*+,)- .%'#"/#-0%'#1"/-230*/"4#567668!$!14#%00$:+#>=0+:$

K:L$MIJN(N?I=CI$<.O./.01*$:KPL$<)O,=$M,M(//.O,$Q(J)NR$ASL$C.OM)?I-$?.O.0J,M<1R$KTSL$M.N(?J.=$IO(NN(.=$?.O.0J,M<1R$J,-UNC.JIL$J,-(.O(C
NC.JIR$?<I$0JII=$C(JC/I$JIMJINI=?N$0J.NN$?)O.J$Q./)OI&

96"<

!"#$%&'(#)*+#$%&$,(#-.)/01#234#53*/& )*+#16347&
1849-9)5#:)7-0*71"=

J*&+*,KL("('%-6')(7%/@.*",(D%0'/5%.$'((%.A&(7%/0%7"#,*.6'(7%9:;9<=(%&2"$418'

7*&"&3$*(%&2"$418'

!"#

!$#

7*&"&3$*($")(9:;9<=(%&2"$418'!"#!$%&'()*+,+-).-'/0102034#(0+,-*0.')1,++,0.'-0103*5(/4!%#%#!&?4+2%$#2%)&')9'-4*4#2#(%=%2,')9'*4-+%+24&2'1)")=)$,'
94#23-4+'9)-'*#2%4&2+'>%21'=3&$'5#&54-'(#+4/')&'
5)"*324/'2)")$-#*1,'%"#$4+!1>)]R+),-#&.-( /+)>?)(JI)S>?(E%J#?>;)56558>1,3 71,"2 /<(!?(:&)'4$@$ AB!CDBA("?(=$@1E&(FGHACA(#?(I14$8, FAJA=A($! :,(7K&(C&"K(7&4.(G"3,*,2.(:,%-&4$*?(:,(7K&(C&"K(7&4.?(L&'4"$E" M$31*4.(,N(C')&3$*(O3&'"3'%?(F.1%K1(D"&#'8%&4.?(P$-$"# Q8$)1$4'(%3K,,*(,N(C')&3$*(O3&'"3'%?(F.1%K1(D"&#'8%&4.?(P$-$"$ O3K,,*(,N(!"N,8E$4&,"($")(H$4$(O3&'"3'%?(R$2$%$@&(D"&#'8%&4.?(P$-$"%"

97

L*.('"*+7%*,D%5(.$/D7!(#")*(+'!TM4!'*)&)%-&i@;$.,'0).0-(JK-%&'%0).%!
!
!

!-L&-%&'%0).%9)/-'1)&+-.&%
9)/-'+/),+.,'1("&"0"<%IP'/+.$&-%')1)(&!"#$%&'
T.&()0<)%%'0"((-<)&+".'0"-33+0+-.&%%% @T::%J'"3'+/),-'3-)&$(-%'#-&7--.'&7"'%0).%Z'! C'/-)%$(-'"3'(-1-)&)#+<+&='! !-3<-0&+".'"3'#"&2'*-,(--'"3'0"((-<)&+".').*'),(--/-.&'#-&7--.'/-)%$(-/-.&%

!"'#

!"#
)*+

,*-./01.

!"(
2**-3

$%%&
456.77.813#$%&'($)*$+',-.$$/01*$2(.3(4567*$-6$'8$0$!"#$%&%'( 9"":##$;(($<0;0*$/5$20=0$)*+,-$.%/.*+01#*+2345%!

L(*,%=::7%/0%"5*#(%0(*.6'(7%@(.?((,%.?/%7-*,7%0/'%D"00('(,.%
M6*,."N*."/,%+()(+7%

!"$&!
!"$'(

!"#$%

%%

!"#$%&#:/5&*'"7/,%/0%=::7%*5/,#%.$'((%7"#,*.6'(7!"##$%&'(#)*+(),(-&'()*+),)-.+/'0),,121+-1

-

%#

98

,9-+0*;01#32#810285*011#32#73:353/<#73#;6)4);704-=0#;)*;04#)*+#)+90410#090*71!0 >?'.&5@' 2*$A@5,B@' C*$;(D'E'.&5$;*$2(@5E','$;*$;(F','$<*$<(3BF' <*$G5H(,5E'$;*$I3',(6($C*$J?'35 C0$2'?H-65K$L-.(H'HK-M)'.-F$J,'?5H?$)5(N.E$456&$O5?H'6B@-.$JHK8BF5H?$<(N(8(?5K'8$)-665 GB,D-@$P-'6B@-.$Q(@$1@-F5K65(H$(Q$1@5,'@E$)@'5H$2-6'.6'65K$O56-.0$1&E.$>H? OK5$2-F0$9"9R$
S'KK-N6-F$(H$+B8E$9!*$9"9RT
90 J3B.&5,' ;*$A@5,B@' C*$='.B,'6.B L*$;',-7'4' C*$G5H(,5E'$;0$<(N(8(?EMD'.-F$@'F5(,5K$Q-'6B@-.$Q(@$N@-F5K65(H$(Q$N'@(65F$?8'HF$K'HK-@$,'85?H'HKE$?@'F-$5H$,'?H-65K$@-.(H'HK-$5,'?-.0$2AU2A0$9"9R$AN@$9"0$F(5V$!"0!""WX.!"RRYM"9RM"!"ZYM"0$>NBD '&-'F$(Q$N@5H60$12J[V$RW"W:!\Y0
R0 G5H(,5E'$;*$A@5,B@' C*$<'H'3'$;*$]&'H$^=*$;'D'6' =*$257BH($O*$U(4F& GP2*$=''3BN GA*$/5',$];*$]&'5$]O*$G?$;C0$<&@--MF5,-H.5(H'8$6(N(8(?5K'8$@'F5(?-H(,5K. (Q$-N5F-@,'8$?@(46&$Q'K6(@$@-K-N6(@$[-8!:$'HF$/Z\ZL$,B6'65(H$.BD6EN-.$(H$K(,NB6-F$6(,(?@'N&E$5,'?-.$(Q$8BH?$K'HK-@$N'65-H6.0$](,NB6 2-6&(F.$1@(?@',.$)5(,-F0$9"9R$+BH_9R`V!"W\YY0$F(5V$!"0!"!`Xa0K,ND09"9R0!"W\YY0$>NBD 9"9R$AN@$!R0$12J[V$RW!YZ``Z0
Y0 /-$b]*$A@5,B@' C*$G5H(,5E'$;*$;(F','$<*$2(@5E','$<0$]'H$1-@.5.6-H6$C(,(8(?E$P-'6B@-.$]'N6B@-$2(@-$JH6@5H.5K$JHQ(@,'65(H$'D(B6$<B,(@.$Q@(,$!ZPMP8B(@(F-(cE?8BK(.-$1(.56@(H$>,5..5(H$<(,(?@'N&EX](,NB6-F$<(,(?@'N&E$J,'?-.$(Q$C-'F$'HF$G-K3$]'HK-@$1'65-H6.d$2-6'D(856-.0$9"99$IK6$!Y_!9S!"TV:W90$F(5V$!"0RR:"X,-6'D(!9!"":W90$12J[V$R`9:\ZWY_$12]J[V$12]:`!"Z\R0\0 G5H(,5E'$;*$A@5,B@' C*$]&'H$^=*$-6$'80$L(DB.6$@'F5(?-H(,5K. 'NN@('K&$6($6&-$5F-H65Q5K'65(H$(Q$>UPL$,B6'65(H.$',(H?$N'65-H6.$456&$GO]/]$Q@(,$
6&@--$F5QQ-@-H6$K(BH6@5-.$B.5H?$6(N(8(?5K'88E$5He'@5'H6$)-665 HB,D-@.0$1/(O IH-0$9"9!_!`S!TV-"9YYR\Y0$F(5V$!"0!RW!Xa(B@H'80N(H-0"9YYR\Y0$`0 G5H(,5E'$;*$A@5,B@' C*$=(.&56'3- <*$-6$'80$OEH-@?5.65K$K(,D5H'65(H$(Q$'$6(N(8(?5K'88E$5He'@5'H6$5,'?5H?$.5?H'6B@-$'HF$'$D5(,'@3-@$Q(@$6&-$'KKB@'6-$
N@-F5K65(H$(Q$.E,N6(,'65K$@'F5'65(H$NH-B,(H565.$D-Q(@-$.6-@-(6'K65K$'D8'65e-$@'F5(6&-@'NE$Q(@$8BH?$K'HK-@V$A$@-6@(.N-K65e-$'H'8E.5.0$1/(O IH-0$9"99$+'H$R!_!WS!TV-"9`R9:90$F(5V$!"0!RW!Xa(B@H'80N(H-0"9`R9:90
W0 ;(F','$<*$A@5,B@' C*$O&5@'3'4'$=*$G5H(,5E'$;*$=(.&56'3- <*$O&5(E',' =0$L-8'N.-$N@-F5K6'D5856E$(Q$6(N(8(?5K'8$.5?H'6B@-$(H$N@-6@-'6,-H6$N8'HH5H?$]<$5,'?-.$(Q$.6'?-$J$H(HM.,'88$K-88$8BH?$K'HK-@$N'65-H6.$D-Q(@-$6@-'6,-H6$456&$.6-@-(6'K65K$'D8'65e-$@'F5(6&-@'NE0$<&(@'K ]'HK-@0$9"99$
AB?_!RS!\TV9!!WM9!9`0$
Z0 G5H(,5E'$;*$A@5,B@' C0$C(,(8(?5K'8$@'F5(,5K.$'H'8E.5.$Q(@$N@(?H(.65K$N@-F5K65(H$5H$8BH?$K'HK-@$N'65-H6.0$1&E.$2-F0$9"9"$+'H_`:V:"M!""0$F(5V$!"0!"!`Xa0-a,N09"!:0!!0"9`0$>NBD 9"!:$[-K$!`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
E#%D;')%<)E-(#$)&>?(;?(;>;)! ,DD-%W#&(?>J')7^_)%<)E-(#$)&>?(;?(;>;)->&(#$).$#=>$?#<#>=! V$O(;#O>)E#%D;')@($)#&D%;>)()J(-:>)E.-=>$)%$)
D(?#>$?;`)#$)D(-?#@.J(-+)#?)#;).$;.#?(EJ>)<%-)D(?#>$?;)#$)D%%-)@%$=#?#%$)%<

99

B#$#44,+)*1"*616&6',4&'#*%#$/61(13&#4*%/,%*,&-*%/#*$2++#6%*&6',4&'#*#C,0&6,%&164 1+*,(%#+6,%&'#*,;;+1,$/#4*%1*1'#+$10#*%/#*&442#4!"#$%&'%()*+($#,*-("*."%+/"0*&/)&#"*1%'#1%=

O$*.%"7%.$(%!(77"*,%",D(P%./%-$*'*-.('"N(%.65/'%$(.('/#(,(".AQ
HI4%JK%:+01-+' @K%L(0%M48%2N2NO
aM> />;;#($ &(?-#W %< >(@M O%W>J %< ( U9 #&(:> !"#$ %$ &' @($ E> :#O>$ E'
GM>->#!! $% &% '% (! " #"! $% &% '% (##! $% &% '% (!!+",#- #$ %$ &$ ( )#!" $% &% '% (#"" $% &% '% (##" $% &% '% (*$+ #$ %$ & , ! #$ %$ &*# % *% & *& ' ()#!# $% &% '% (#"# $% &% '% (### $% &% '% (%*$+ #$ %$ &*# % *% & *& ' (, ! #$ %$ & $CB'DB'% E *F C < D < % , (B'6. 1# 2# G +%'&2-'5)$%%+).'3+<&-('7+&2')'9M'8B').*'i'*-."&-%'0".>"<$&+".

!" -(!# " &:# !$ !" $ !# $ !$ _(2"*3%-(3$51:#-(&:#(%/"*#($"L$:9&6,$3(12(%-($&5/(12(R/"5/(*$>*$3$:%3(/$%$*1L$:$"%7(12(&(%,+1*&% ' ( &% ( ) _(!$33"&:(":#$`(4:,+K$*(12(:$L&%"9$($"L$:9&6,$3(12(%>-.//012 023.4$ 5)3445%267(86%&"%9"*(':"
4%6&8";68:6&"("8<7%5"<=6&>"?<(-5(86'"9<&'86%&=!"#$%&'()*+,"(&-"."#/(--0*+

1"#$%&)*2+* + ,- #., / '0
B+1."*./%(B-%-+#/&(C--,.)A?4&"0)&$%,.$3+,30"&*+,* + ,- #., 1 '0&9!U6b)D(?#>$?;)!c76)a7QG>#:M?>=)@%$?-(;?Q>$M($@>=)&(:$>?#@)->;%$($@>)#&(:>;8)GM%)M(O>)TS)G>->)@M%;>$)<%-)@(J@.J(?#$:)#&(:>)<>(?.->;)($=)@%$;?-.@?#$:)()J#:M?):-(=#>$?)
E%%;?#$:)&(@M#$>)&%=>J)<%-)#=>$?#<#@(?#%$)%<)D-#&(-')@($@>-);#?>;I!aM>)D-%D%;>=)&%=>J)(@M#>O>=)M#:M>-),dR;)%<)6Iee)($=)6Icc)#$)()?-(#$#$:)($=)?>;?+)->;D>@?#O>J'!aM>)D-%D%;>=)(DD-%(@M)@%.J=)M(O>)(
D%?>$?#(J)<%-)#=>$?#<'#$:)D-#&(-')@($@>-);#?>;+)E.? #?)M(;)(J->(=')E>>$ #&D-%O>=I&#

100

@"''%#2$5A8! !"#$"%&"'"#()*(#()"#*$&%&(+#,-#(,.,%,/+#-,0#1)*0*1("0&2&3/#1*31"0#/",4"(0+#5*6#-,7389#! :)"#*$&%&(+#&6#";("38"8#(,#.0"8&1(&,3#,-#.*(&"3(6<#.0,/0*3,6"6A".(01'%4&,"(,8(3,EE'"4%(S&!

101

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Singularity theory and its applications to strongly
convex multiobjective optimization problems
Shunsuke Ichiki
Department of Mathematical and Computing Science, School of Computing, Tokyo
Institute of Technology, Japan
A multiobjective optimization problem is a problem to optimize multiple objectives,
such as cost, quality, safety and environmental impact in the industrial world. In thismini-course, I would like to introduce theoretical applications of “singularity theory ofdifferentiable mappings”, which is a branch of geometry, to strongly convex multiobjective optimization problems.
For this purpose, we first introduce some of basic notions of singularity theory. We
also discuss a result called a “parametric transversality theorem”, which is an importantand fundamental tool in singularity theory for investigating generic mappings. Then,
as an application, we give a transversality theorem on linear perturbations. Next, we
explain some basic notions of multiobjective optimization and introduce a property of
the Pareto set (i.e. the set of optimal solutions) of a strongly convex multiobjective
optimization problem from the viewpoint of topology. Finally, based on them, we introduce theoretical applications of singularity theory to strongly convex multiobjective
optimization problems.

103

104

105

106

107

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Explanatory Model Analysis
Przemyslaw Biecek
MI2 Data Lab, Faculty of Mathematics and Information Science,Warsaw University of Technology, Poland
Shapley values currently stand as the most widely employed technique for conducting
Explanatory Model Analysis (EMA) and achieving Explainable Artificial Intelligence(XAI). Ongoing efforts are focused on crafting modifications and extensions to adapt
this method to address the diverse challenges posed by a wide array of applications.
In this presentation, I will illustrate instances where Shapley values, and by extension,techniques utilized in explainable artificial intelligence, prove effective in distinguishingmodels exhibiting distinct behaviors, even if their performance appears identical atfirst glance. Subsequently, I will present a proposal for an iterative model analysis
process utilizing Shapley values. Drawing inspiration from Rashomon perspectives, thisapproach, termed Shapley Lenses, provides a more nuanced perspective on predictivemodels. The insights derived from predictive models can then be leveraged to constructsubsequent iterations of models with enhanced interpretability.

109

Explanatory Model
Analysis
aka Explainable AI
Przemysław Biecek/ˈpʂɛ.mɛk/

WORKSHOP on Mathematics for Industry 2023

Przemyslaw Biecek

September 25, 2023

Why?

Hype for AI is growing

https://www.slideshare.net/jaypod/digitaltransformation50soundbites/19-Data_is_the_new_oilClive
https://www.newworldai.com/forget-the-hype-what-every-business-leader-needs-to-know-about-artiÞcial-intelligence-now/

110but AI is broken

https://incidentdatabase.ai/

but AI is broken

https://www.slideshare.net/jaypod/digitaltransformation50soundbites/19-Data_is_the_new_oilClive
https://www.newworldai.com/forget-the-hype-what-every-business-leader-needs-to-know-about-artiÞcial-intelligence-now/
https://medium.com/@Joy.Buolamwini/response-racial-and-gender-bias-in-amazon-rekognition-commercial-ai-system-for-analyzing-faces-a289222eecedhttps://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing

but AI is broken

https://www.slideshare.net/jaypod/digitaltransformation50soundbites/19-Data_is_the_new_oilClive
https://www.newworldai.com/forget-the-hype-what-every-business-leader-needs-to-know-about-artiÞcial-intelligence-now/
https://medium.com/@Joy.Buolamwini/response-racial-and-gender-bias-in-amazon-rekognition-commercial-ai-system-for-analyzing-faces-a289222eecedhttps://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing

111

but AI is broken

https://www.slideshare.net/jaypod/digitaltransformation50soundbites/19-Data_is_the_new_oilClive
https://www.newworldai.com/forget-the-hype-what-every-business-leader-needs-to-know-about-artiÞcial-intelligence-now/
https://medium.com/@Joy.Buolamwini/response-racial-and-gender-bias-in-amazon-rekognition-commercial-ai-system-for-analyzing-faces-a289222eecedhttps://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing

but AI is broken

https://www.slideshare.net/jaypod/digitaltransformation50soundbites/19-Data_is_the_new_oilClive
https://www.newworldai.com/forget-the-hype-what-every-business-leader-needs-to-know-about-artiÞcial-intelligence-now/
https://medium.com/@Joy.Buolamwini/response-racial-and-gender-bias-in-amazon-rekognition-commercial-ai-system-for-analyzing-faces-a289222eecedhttps://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing

so, we are here to fix AI
We develop methods, tools and processes
for responsible machine learning.

112

How?

Shapley values

Problem A:
Set S with n players cooperating in a game can earn a reward v(S).
How to divide this reward fairly among the players?

Shapley values

Shapley values

Shapley, Lloyd S. A Value for n-Person Games. Princeton University Press. 1952

113

Shapley values

Shapley values

Shapley, Lloyd S. A Value for n-Person Games. Princeton University Press. 1952

Shapley values for ML models

Problem B:
In machine learning, we train a function f(x) : Rp→R that
calculates predictions based on p variables.
How to quantify the effect of each variable on the final
prediction?

Shapley values for ML models

Erik Strumbelj, Igor Kononenko, Marko Robnik-Sikonja. Explaining instance classifications withinteractions of subsets of feature values. Data & Knowledge Engineering, 2009114

Shapley values for ML models
Example:We have a complex predictive modelf(x) that predicts the probability ofsurviving the Titanic disaster.
How do we calculate contributions ofindividual variables to the prediction
for a single passenger.
Here: an eight-year-old boy travelling
1st class.

Explanatory Model Analysis. Przemyslaw Biecek, Tomasz Burzykowski. 2021. CRC. https://ema.drwhy.ai

Shapley values for ML models
Example:We have a complex predictive modelf(x) that predicts the probability ofsurviving the Titanic disaster.
How do we calculate contributions ofindividual variables to the prediction
for a single passenger.
Here: an eight-year-old boy travelling
1st class.

Explanatory Model Analysis. Przemyslaw Biecek, Tomasz Burzykowski. 2021. CRC. https://ema.drwhy.ai

Shapley values for ML models
Example:We have a complex predictive modelf(x) that predicts the probability ofsurviving the Titanic disaster.
How do we calculate contributions ofindividual variables to the prediction
for a single passenger.
Here: an eight-year-old boy travelling
1st class.

Explanatory Model Analysis. Przemyslaw Biecek, Tomasz Burzykowski. 2021. CRC. https://ema.drwhy.ai

115

Shapley values for ML models
Example:We have a complex predictive modelf(x) that predicts the probability ofsurviving the Titanic disaster.
How do we calculate contributions ofindividual variables to the prediction
for a single passenger.
Here: an eight-year-old boy travelling
1st class.

Explanatory Model Analysis. Przemyslaw Biecek, Tomasz Burzykowski. 2021. CRC. https://ema.drwhy.ai

SHAP as an unification of LIME, DeepLIFT,
Layer-Wise Relevance Propagation

Lundberg, Scott M, Su-In Lee. A Unified Approach to Interpreting Model Predictions. NeurIPS. 2017

TreeSHAP as quick SHAP for Tree Ensembles

Lundberg, Scott M,Gabriel G. Erion, Su-In Lee. Consistent Individualized Feature Attribution
for Tree Ensembles. NeurIPS. 2019116

SurvSHAP(t) for survival models

Problem C:
In survival modelling, the model output isa survival function S(t).
How to quantify the effect of each variable on the final
prediction (which is a function)?

SurvSHAP(t) for survival models

Mateusz Krzyziński, Mikołaj Spytek, Hubert Baniecki, Przemysław Biecek.
SurvSHAP(t): Time-dependent explanations of machine learning survival models. Knowledge-Based Systems. 2023

How to calculate SurvSHAP(t) values
Contribution of variable 𝒅𝒅 in time point 𝒕𝒕 for the patient 𝒙𝒙:

Local variable importance of variable 𝒅𝒅 for the patient 𝒙𝒙:

Mateusz Krzyziński, Mikołaj Spytek, Hubert Baniecki, Przemysław Biecek.
SurvSHAP(t): Time-dependent explanations of machine learning survival models. Knowledge-Based Systems. 2023

117

SurvSHAP(t) can detect time-dependent
variable effects

Mateusz Krzyziński, Mikołaj Spytek, Hubert Baniecki, Przemysław Biecek.
SurvSHAP(t): Time-dependent explanations of machine learning survival models. Knowledge-Based Systems. 2023

SurvSHAP(t) show global variable importanceMateusz Krzyziński, Mikołaj Spytek, Hubert Baniecki, Przemysław Biecek.
SurvSHAP(t): Time-dependent explanations of machine learning survival models. Knowledge-Based Systems. 2023

SurvSHAP(t) shows the dependenceof variable attributions on its values

Mateusz Krzyziński, Mikołaj Spytek, Hubert Baniecki, Przemysław Biecek.
SurvSHAP(t): Time-dependent explanations of machine learning survival models. Knowledge-Based Systems. 2023

118

SurvSHAP(t) can be analyzed using
functional data analysis technique

Mateusz Krzyziński, Mikołaj Spytek, Hubert Baniecki, Przemysław Biecek.
SurvSHAP(t): Time-dependent explanations of machine learning survival models. Knowledge-Based Systems. 2023

Application of SurvSHAP(t)Hubert Baniecki, Bartlomiej Sobieski, Przemyslaw Bombinski, Patryk Szatkowski, Przemyslaw Biecek. Hospital Lengthof Stay Prediction Based on Multi-modal Data towards Trustworthy Human-AI Collaboration in Radiomics. AIME. 2023

Application of SurvSHAP(t)Hubert Baniecki, Bartlomiej Sobieski, Przemyslaw Bombinski, Patryk Szatkowski, Przemyslaw Biecek. Hospital Lengthof Stay Prediction Based on Multi-modal Data towards Trustworthy Human-AI Collaboration in Radiomics. AIME. 2023

119survex: open software for XAI+surv

When?

Rashomon set

Leo Breiman. Statistical Modeling: The Two Cultures. Statistical Science. 2001120

Rashomon quartet
Fifty years ago, Anscombe presented the construction of four
datasets covering different relationships but with identical RMSE.
22 years ago, Breiman discussed the concept of the Rashomon set- models with the same fit to the data but different relationships
between the predictor variables with the target variable.

Przemyslaw Biecek, Hubert Baniecki, Mateusz Krzyzinski, Dianne Cook. Performance is not enough:the story told by a Rashomon quartet. 2023 https://arxiv.org/2302.13356

Explanatory Model
ModelAnalysis
Exploration Stack
What is the model prediction
for the selected instance?

How good is the model?
f(x)ROC curve
LIFT, Gain charts
Chapter 16

AUC
RMSE

Which variables contribute to
the selected prediction?

Which variables are importantto the model?

Break Down
SHAP, LIME
Chapters 7, 8, 9, 10

Permutational
Variable Importance
Chapter 17

How does a variable affectthe average prediction?

How does a variableaffect the prediction?
Ceteris Paribus
Chapters 11, 12

Partial Dependence ProﬁleAccumulated Local Effects
Chapters 18, 19

Does the modelﬁt well aroundthe prediction?

Does the modelﬁt well in
general?

Chapter 13

PREDICTION LEVEL

Chapter 20

Przemyslaw Biecek, Tomasz Burzykowski. Explanatory Model Analysis. CRC. 2021
MODEL LEVEL

The process of explanatory model analysis

Michael Bücker, Gero Szepannek, Alicja Gosiewska, Przemyslaw Biecek. Transparency, Auditability and eXplainabilityof Machine Learning Models in Credit Scoring. Journal of the Operational Research Society. 2022

121

EMA process validated with user-studies

Hubert Baniecki, Dariusz Parzych, Przemyslaw Biecek. The grammar of interactive explanatory model analysis.
Data Mining and Knowledge Discovery. 2023

EMA process validated with user-studies

Hubert Baniecki, Dariusz Parzych, Przemyslaw Biecek. The grammar of interactive explanatory model analysis.
Data Mining and Knowledge Discovery. 2023

Who?

122

MI2.AI is here to fix AI

PapersSoftware

Books

123

TeachingWORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Linear instability of Prandtl spirals
Tomasz Cieślak
Institute of Mathematics, Polish Academy of Sciences, Poland
We review a recent result with P.Kokocki and W.Ożański stating that the union of
three or more uniformly distributed Prandtl spirals is linearly unstable as a solution to
the Birkhoff-Rott equation. First, a linearization of the Birkhoff-Rott equation aroundthe Prandtl spirals is found. Next, a perturbation leading to the instability is shown.
Notice that, unlike for the flat sheet, the unstable modes grow only algebraically intime. In our talk we partially answer the question of Helmholtz from his famous 1868
paper on discontinuous flows.

125

126

127

128

129

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Pseudospheres from singularity theory view-pointwith a classification of 2-soliton surfaces
Toshizumi Fukui
Department of Mathematics, Saitama University, Japan(joint work with Yutaro Kabata)
We discuss pseudospheres in the Euclidean 3-space with taking care about their singularity types and Backlünd transformations. We investigate a classification of 2-soliton
surfaces by noting how the ridge lines appear.

131

Pseudospheres fromsingularity theory view pointwith a classificationof 2-soliton surfaces(j/w with Yutaro Kabata)Toshi Fukui (Saitama University)
16:45–17:15, 26 September, 2023
Workshop for Mathmatics for Industry
25–29 September, 2023
Warsaw University of Technology

Surfaces in R3
φ : R2 −→ M = φ(R2 ) ⊂ R3 , C ∞E = φu , φu , F = φu , φv , G = φv , φv L = φuu , ν, M = φuv , ν, N = φvv , ν

where ν is a unit normal.
The first fundamental form

I = E du 2 + 2F du dv + G dv 2
The second fundamental form
II = L du 2 + 2M du dv + N dv 2
2 / 16

Chebyshev’ net
A pseudosphere is a surface with constantnegative Gauss curvatures. We can assume thatthey have Gauss curvature −1 up to similarity
transformations.
For a surface with K = −1, we can take theasymptotic coordinate (u, v ) with the following
fundamental forms:
I =du 2 + 2 cos ϕ du dv + dv 2
II =2 sin ϕ du dv
where ϕ is the asymptotic angle.
Gauss-Coddazi equation becomes sine-Gordon
equation:
ϕuv = sin ϕ
3 / 16132

Curvature coordinate
The curvature coordinate is given by
x=

u+v
,
2

y=

u−v
.
2

The fundamental forms are
I = cos2 ϕ2 dx 2 + sin2

ϕ 2
dy
2

1
II = sin ϕ (dx 2 − dy 2 )2The principal curvatures areϕϕtan , and − cot2
2
4 / 16

Ridge and flecnodal
Let vi denote a principal vector of a surface andlet κi denote the corresponding principal
curvature of a surface.
A point P on a surface is vi -ridge if vi κi (P) = 0.
A point P on a surface is flecnodal if there is aline with at least 4 point contact with thesurface at P.

5 / 16

1. The level sets of ϕ, κ1 and κ2 contaning P are
equal.
2. The differentials of the principal curvatures
are given as follows:ϕx
ϕy∂x κ1 =,∂y κ1 =,
1 + cos ϕ
1 + cos ϕϕx
ϕy,∂y κ2 =.
∂x κ 2 =−1 + cos ϕ−1 + cos ϕSo ∂x -ridge (resp. ∂y -ridge) is given by ϕx = 0(resp. ϕy = 0). (A level of ϕ has a horizontal (orvertical) tangent.)
Flecnodal point on pseudosphere is given by
ϕu ϕv = 0. (i.e., A level of ϕ has a diagonal (oranti-diagonal) tangent.)

6 / 16133

Backlünd transformation
We say ϕ̃ is Backlünd transformation of ϕ if
 ϕ̃ + ϕ 2u= λ sinϕ̃ − ϕ,2

 ϕ̃ − ϕ 2v= λ−1 sinϕ̃ + ϕ. (1)2where λ = tan θ/2. θ is in the next sheet.
If ϕ is a solution of sine-Gordon equation, so is ϕ̃.
BT{sol. of sine-Gordon} −→ {sol. of sine-Gordon}

7 / 16

Geometric BT
We sayM −→ M,p −→ p̃,is gemetric BT, if
• The line p p̃ is in Tp M and also in Tp̃ M.
• d(p, p̃) is constant (= r ).
• the unit normals ν p and ν̃ p̃ has a constantangle θ, that is ν p , ν̃ p̃  = cos θ.
Geometric BT between K = −1 surfaces is givenby cos ϕ̃/2
sin ϕ̃/2 
φ̃ = φ + r
φx +φy , r = sin θcos ϕ/2
sin ϕ/2and it preserves Chebyshev’s nets.

8 / 16

Bianchi’s permutability
If ϕi (i = 1, 2) satisfies
ϕ + ϕϕi − ϕi,= λi sin22uand ϕ̃ satisfies(λ2 − λ1 ) tan

ϕ − ϕ
i

2

v

= λ−1
i sin

ϕi + ϕ,2

ϕ̃ − ϕϕ2 − ϕ 1= (λ2 + λ1 ) tan
,
4
4

then
 ϕ̃ + ϕ 
ϕ̃ − ϕ1
1,= λ2 sin22u ϕ̃ + ϕ 
ϕ̃ − ϕ2
2,= λ1 sin22u ϕ̃ − ϕ 12

 ϕ̃ − ϕ 

v

2

2

134

v

= λ−1
2 sin

ϕ̃ + ϕ1
.
2

= λ−1
1 sin

ϕ̃ + ϕ2.
2
9 / 16

Soliton
0-soliton
ϕ=0

BT 

1-solitonBT 

2-solitonξ = λi u + λ−1i v
i
ξ1 −ξ2 λ1 +λ2 sinh 2·
ϕλ1 ,λ2 = 4 tan
ξ1 +ξ2λ2 −λ1ϕλ = 4 tan−1 (λu + λ−1 v )

−1

line

cosh

2

Beltrami’s pseudosphere
Dini’s pseudosphere2-soliton surfaces

10 / 16

Singular locus of φ
Let φ : R2 → R3 be a Chevyshev net for apseudosphere with K = −1. Let ϕ denote theasymptotic angle. ThenI =du 2 + 2 cos ϕ du dv + dv 2
II =2 sin ϕ du dv
Remark that the singular locus of φ is defined byΣ : sin ϕ = 0,i.e., ϕ = kπ, k ∈ Z.

For 2-soliton surface, we have k = 0, ±1.
11 / 16

Criteria of singularities
Let C denote the curvature line through P whose
principal direction is null direction at P.
1. Assume that ϕ is nonsingular at P, i.e., thesingular locus of φ is nonsingular at P.
1.1 φ is cuspidal edge at P if and only if Σ and C
intersect transversely at P.
1.2 φ is swallowtail at P if and only if Σ has
2-point contact with C at P.

2. Assume that ϕ has a Morse singularity at P.
2.1 φ is cuspidal beaks at P if and only if the
Hessian of ϕ is positive.
2.2 φ is cuspidal lips at P if and only if the Hessianof ϕ is negative.
12 / 16135

Flecnodal and ridge on a
2-soliton surface
On pseudospheres, we have∂u -flecnodal line (ϕu = 0),∂v -flecnodal line (ϕv = 0),∂x -ridge line (ϕx = 0),∂y -ridge line (ϕy = 0)
and, on 2-soliton surfaces, they are
λ2 λ1 λ2 + λ−1
λ2 − λ−1cosh ξ22
2= ,,, respectively.
−1 ,cosh ξ1
λ1 λ2 λ1 + λ1
λ1 − λ−11Here ξi = λi u + v /λi , i = 1, 2.
13 / 16

When λ2 → λ1 = λ,ϕλ,λ = lim
ϕλ,λ′ = 4 tan−1
′
λ →λ

−η
,
cosh ξ

where ξ = λu + λ−1 v + c and η = λu − λ−1 v .
The ∂u -flecnodal, ∂v -flecnodal, ∂x -ridge and ∂y -ridge
are defined byη tanh ξ = 1, −1,λ − λ−1 λ + λ−1,, respectively.
λ + λ−1 λ − λ−1

14 / 16Classification of 2-soliton
The result in this section should compare theclassification of 2-soliton surfaces (Popov). Theyshow four types for generic 2-soliton surfaces.
The correspondence between their classificationand our results is summarized as follows:
Type λ1 λ2
1
+
2
+
3
−
4
−

µ
+
−
+
−

flecnodal
exist
exist
not exists
not exists∂x -ridge∂y -ridge
exist
exist
exist
not exist
not exist not exist
not exist
existµ = (λ21 − 1)(λ22 − 1).
15 / 16136

Breather surfaces
For λ ∈ C, with Re λ = 0, Im λ = 0, we havesin Im ξ 
ϕλ,λ̄ = − 4 tan−1 cot arg λ ·cosh Re ξ

where ξ = λu + v /λ.
The ∂u -flecnodal, ∂v -flecnodal, ∂x -ridge and ∂y -ridge
are defined by(tanh Re ξ)(tan Im ξ)|λ| − |λ|−1 |λ| + |λ|−1= 1, −1,,,
tan arg λ|λ| + |λ|−1 |λ| − |λ|−1respectively.
16 / 16137

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Formation of nanostructured functional particles
with the spray-drying method
Leon Gradoń
Faculty of Chemical and Process Engineering,
Warsaw University of Technology, Poland
The structure of matter, on both an atomic and macroscopic scale, is a result of theinterplay between the requirements of the physical forces operating between the individual parts and the mathematical requirements of space-filling. Nanoparticles withwell-defined chemical composition can act as a building block for the construction offunctional structures, such as highly ordered aggregates, as well as porous and hollow
aggregates. A spray drying technique is used for the production of crystal-like structures with nanoparticle building blocks. When spray-drying uniform spherical particles
tightly packed aggregates with either simple or broken symmetry were formed using
polystyrene particles with varied zeta potential as templates, it is also possible to form
highly ordered porous and hollow aggregates from inorganic colloidal particles potentially useful for controlled drug delivery and catalysis. The process by which organizedmesoporous silica particles are formed by the spray-drying method was examined using
elementary laws of topology.

139

Formation of nanostructured functional
particles with the spray-drying method
Leon Gradon
Faculty of Chemical and Process Engineering,
Warsaw University of Technology Warsaw, Poland
Stanisław Janeczko
Faculty of Mathematics and Information Sciences
Warsaw University of Technology Warsaw, Poland
Ratna Balgis, Takahashi Ogi, Kikuo Okuyama
Department of Chemical Engineering, Hiroshima University,
Higashi-Hiroshima, Japan

Contents
1. Introduction
2. Principles of self-assembly
3. Shapes of the structures
4. Examples of nanostructures applications
5. Principle of spray-drying process
6. Examples of produced templates

7. Topographical structures for challenging aspects of nanocatalysis
8. Conclusions“There is plenty of room at the bottom”
Richard P. Feynman(there is a room for great development even in the microscopic world)

140

The structure of matter, on both an atomic and macroscopic scale, is a
result of the interplay between the requirements of the physical forces
operating between the individual parts and the mathematicalrequirements of space filling.

Types of self-assembly
Static self-assembly (S)S – involves systems that are at global or local equilibrium and do not dissipate energy

Examples of static self-assembly(A) Crystal structure of a ribosome(B) Self-assembled peptideamphiphile nanofibers.
(C) An array of millimetersized polymeric plates
assembled at a water/perfluorodecalin interface bycapillary interactions.
(D) Thin film of a nematic liquid crystal on an isotropic
substrate.
(E) Micrometersized metallic polyhedra folded from planar
substrates.
(F) A three-dimensional aggregate of micrometer plates
assembled by capillary forces.

141

Dynamic self-assembly (D)D – interaction responsible for the formation of structures or pattern between components only occur if thesystem is dissipating energy

Examples of dynamic self-assembly(A) An optical micrograph of a cell withfluorescently labeled cytoskeleton and nucleus;
microtubules (~24 nm in diameter) are colored red.
(B) Reaction-diffusion waves in a Belousov-Zabatinski
reaction in a 3.5-inch Petri dish.
(C) A simple aggregate of three millimeter-sized, rotating,magnetized disks interacting with one another via
vortex-vortex interactions.
(D) A school of fish.
(E) Concentric rings formed by charged metallic beads
1 mm in diameter rolling in circular paths on adielectric support.
(F) Convection cells formed above a micropatterned
metallic support.
The distance between the centers of the cells is ~2 mm.

Using shape for self-assembly
Major milestones towards the goal of self-assembly:
1) Making the building blocks.
2) Understanding and controlling the interactions.
3) Predicting the consequence of many components interacting in a prescribed
environment.
4) Identify components and interactions that will organize to form a desired
product (reverse self-assembly).
5) Knowing how to use self-assembly.

142Templates
A brute force approach to create nearly arbitral shapes uses templates.
Template is a sacrificial mold in which material is grown or deposited, e.g.
micelles, membrane, colloid crystals, zeolites, and block copolymers.

Instabilities
This approach aims to create a highly symmetric yet metastable structure(spherical colloid coated with a metal).
Under the stimulus the structure “relaxes” toward one of its ground states bybreaking its own symmetry, e.g. stimulus heat, shell devotes leading to theformation of a lower symmetry, stimulus-stretch metastable conformation foldinto functional shape (proteins).

The reciprocity in material technology

means

processing

goal

structureproperties

cause and effect143

performancePrinciple of spray-drying process

Principles of spheres arrangementsin the spray-drying process144

Close-packing of spheres in Euclidean space:{S} = {(S1,p1...(SN ,pN )}Two spheres (Si, pi) and (Sj, pj) of radius r are in contact, i.e.:

dist (pi, pj) = 2r
Cluster of spheres is weakly tetrahedral, T, if for each sphere(Si1, pi1) there exist three spheres (Si2, pi2), (Si3, pi3) and (Si4, pi4).
Such the distancedist (Sik, pil) = 2r

if 1≤ k ,l ≤ 4

Tetrahedral nano-cluster(cluster which consists of tetrahedrals)
For every two tetrahedra Til ,Tik there exist an ordered chain:
That Ti n ,Tin+1

{T }
il

have common face, n = 1 … k -1

• Tetrahedron is a basic unit of the tight packing by equal spheres.

•

Distortion associated with tetrahedral packing.

•13 spheres icosahedron have small distortion.

•

12 spheres arranged symmetrically around one sphere are notpacked in perfectly way.

•

Distance a between spheres: a > 2r

• Elementary property of icosahedron gives a relation:()

a = 8r / 10+ 2 5
145

1/ 2

l =1...kCompact packing of spheres

Electric
furnace
Diffusion
dryer

Filterholder

MFC Pump

CycloneCooling
SystemFilter
MFC
7
8

N2

6

5

4

3

2
1

Ultrasonic Nebulizer

Organization of spheres in the droplet

146

Organization of spheres in the droplet

Examples of produced templates

147

Cell structure on the plane
P = {P1,…..PN} – cell centers on R2

dk(x) = d(x , Pk) – Euclidean distance function from Pk
fk : R2→ R – cell structure function
Global competition squared distancefunction:





2
2
dˆ ( x ) = min f1 ( x )d 1 ( x ),..... f N ( x )d N ( x )148

Conflict sets arrangements and the intercells patterns
System of competition organizing centers

P = {P1,P2…..Pn} = {(P1 , f1) , (P2 , f2) .... (Pn , fn)}
Cells corresponding to the system

CPi = {xR2 : k(1...n) fi(x) di2(x)  fk(x) dk2(x)}Interface curve (generalized Voroni diagram)

VP = { xR2 : ifj S that kfid2(x, pi) =fj d2(x, pj) = fj d2(x, pj)  fk d2(x, pk)}
assumption: fi  1 , i = 1,…N

Euler formula

Euler formula

149

150

Hierarchical organization of particles of different diameterson the surface of sphere

SiO2 Nanostructuration using Ultrasonic SD
Two components
Gas flow: 1.0 L/min, M115/16 is the mass ratio of 115/16 nm SiO2 particles
M115/16 = 6

M115/16 = 4

500 nmM115/16 = 10500 nmM115/16 = 12500 nm

5.0 μm500
nm100
Smooth surface

500 nm

5.0 nmμm
500
nm
5.0 μm
5.0 nm
µm
100
500
500
100
100
nm
Single layer of small particle 115 nm particle are incontact with each other
When the mass ratio of silica particles (M115/16) was increased:

the number of small particles layer decreases
the particles surface changed from smooth to rough151

SiO2 Nanostructuration using Ultrasonic SD
Three components
Mass ratio of SiO2 particles 16 nm : 115 nm : 360 nm
1:2:2

1:3:3

1:2:1

1:5:5

500 nm

500 nm

500 nm

500 nm

5.0 µm500
100
nm

5.0 µm
100
nm
500
nm

5.0 µm100
nm
500

5.0 µm500
100
nm

360 nm particles were surrounded by 115 nm particles
When the mass ratio of large particles was increased,surface morphology changed from smooth to concave-convex

Conclusions•

The aerosol assisted spray-drying process is an useful method for production of developed
and desired space-forms made of nanoparticles.

•

Mesoporous nanostructured particles were produced using PSL particles as a templatematerial for organizing nanoparticles around them.

•

The composition of the cells on the surface of sphere is described using the concept ofconflict set arrangement.

•

Stationary state of cell configuration on the sphere has equal infinitesimal of cellboundaries in real vortex and they are equal 2p/3.

•

The signs of zeta potential of the template particles and colloid particles used in the spraydrying process define the structure of the final product, which could be either hollow or
porous.

Thank you for your attention☺

152

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

On comparing distributions with imprecise data
Przemyslaw Grzegorzewski
Faculty of Mathematics and Information Science,Warsaw University of Technology, Poland
One of the most fundamental problems in mathematical statistics is the comparisonof two or more distributions that characterize the underlying populations. Classicaltests applied there are constructed with pretty specific assumptions concerning the
distributions, like normality, exponentiality, etc. However, in reality, these assumptions
are often not met. The problem becomes much more difficult when the output of anexperiment consists of data that are imprecise, or vague. There we need a model thatallows us to grasp both aspects of uncertainty that appear in such data: randomness,
associated with the data generation mechanism, and fuzziness, connected with data
imprecision. To cope with this problem Puri and Ralescu (1986) introduced a fuzzy
random variable.
On the other hand, in analyzing fuzzy data from the statistical perspective we
immediately come upon some key obstacles, like the nonlinearity associated with the
fuzzy number arithmetic, the lack of a universally accepted total ranking, the lack ofsuitable probability distribution models, or no limit theorems for random mechanisms
producing fuzzy data which could be directly applied in statistical inference. Therefore,
statistical tests with imprecise data usually cannot be generalized straightforwardly
from their classical prototypes.
We show that some of the aforementioned difficulties in test construction can beovercome by using permutation-based nonparametric procedures. Combining thesewith a distance-based approach or a dominance credibility index gives us some interesting goodness-of-fit and location tests, respectively.

153

154

155

156

157

158

159

160

161

162

163

164

165

166

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Brief Introduction to Topology for Multi-objective
Optimization
Naoki Hamada
Machine Learning Group, KLab Inc.
A broad range of scientific and engineering tasks, including data analysis, productdesign, modeling, planning, and management, can be formulated in multi-objective
optimization problems. Recent developments in convex analysis and data science using topology have brought a new paradigm for solving and analyzing multi-objective
optimization problems. In this talk, several applications of topology to multi-objective
optimization will be presented. We will show how the topology of convex analysis canbe applied to a sparse modeling task, generalizing the regularization path of the elastic net and efficiently tuning its two hyper-parameters simultaneously. To extend thisidea beyond the convexity assumption, we introduce a statistical test using persistenthomology and the Poincaré conjecture whether the hyper-parameter tuning method
works.

167

㼃㻹㻵㻌㻞㻜㻞㻟䈊㻮㼞㼕㼑㼒㻌㻵㼚㼠㼞㼛㼐㼡㼏㼠㼕㼛㼚㻌㼠㼛㻌㼀㼛㼜㼛㼘㼛㼓㼥䈊㼒㼛㼞㻌㻹㼡㼘㼠㼕㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㻻㼜㼠㼕㼙㼕㼦㼍㼠㼕㼛㼚䈊㻺㼍㼛㼗㼕㻌㻴㼍㼙㼍㼐㼍㻌㻔㻷㻸㼍㼎㻌㻵㼚㼏㻚㻕䈊

Copyright (c) KLab Inc. All Rights Reserved.䈊

㻲㼞㼛㼙㻌㼠㼔㼑㻌㼢㼕㼑㼣㼜㼛㼕㼚㼠㻌㼛㼒㻌㼍㼚㻌㼕㼚㼐㼡㼟㼠㼞㼕㼍㼘㻌㼞㼑㼟㼑㼍㼞㼏㼔㼑㼞㻚㻚㻚䈊㻵㻌㼣㼕㼘㼘㻌㼠㼍㼘㼗㻌㼍㼎㼛㼡㼠䈊● 㻾㼑㼘㼍㼠㼕㼛㼚㻌㼎㼑㼠㼣㼑㼑㼚㻌㼙㼛㼎㼕㼘㼑㻌㼓㼍㼙㼑㻌㼕㼚㼐㼡㼟㼠㼞㼥㻌㼍㼚㼐㻌㼙㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟䈊○ 㻴㼛㼣㻌㼏㼡㼠㼠㼕㼚㼓㻙㼑㼐㼓㼑㻌㼠㼑㼏㼔㼟㻌㼍㼞㼑㻌㼍㼜㼜㼘㼕㼑㼐㻌㼠㼛㻌㼓㼍㼙㼑㻌㼜㼞㼛㼐㼡㼏㼠㼟䈊○ 㼃㼔㼥㻌㼙㼛㼎㼕㼘㼑㻌㼓㼍㼙㼑㼟㻌㼍㼞㼑㻌㼙㼍㼠㼠㼑㼞㻌㼒㼛㼞㻌㼙㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟䈊● 㼀㼛㼜㼛㼘㼛㼓㼥㻘㻌㼙㼡㼘㼠㼕㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼛㼜㼠㼕㼙㼕㼦㼍㼕㼠㼛㼚㻌㼍㼚㼐㻌㼟㼜㼍㼞㼟㼑㻌㼙㼛㼐㼑㼘㼕㼚㼓䈊○ 㻭㻌㼚㼑㼣㻌㼙㼍㼠㼔㻌㼠㼔㼑㼛㼞㼥㻌㼐㼑㼢㼑㼘㼛㼜㼑㼐㻌㼕㼚㻌㼏㼛㼘㼘㼍㼎㼛㼞㼍㼠㼕㼛㼚㻌㼣㼕㼠㼔㻌㼍㼏㼍㼐㼑㼙㼕㼍䈊○ 㻭㼚㻌㼍㼜㼜㼘㼕㼏㼍㼠㼕㼛㼚㻌㼠㼛㻌㼙㼛㼎㼕㼘㼑㻌㼓㼍㼙㼑㼟䈊2© KLab Inc.

㻹㼛㼎㼕㼘㼑㻌㻳㼍㼙㼑㻌㻵㼚㼐㼡㼟㼠㼞㼥䈊㼍㼚㼐㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟䈊3© KLab Inc.

168㻯㼛㼙㼜㼍㼚㼥㻌㻻㼢㼑㼞㼢㼕㼑㼣䈊㻯㼛㼙㼜㼍㼚㼥㻌㻺㼍㼙㼑䈊㻷㻸㼍㼎㻌㻵㼚㼏㻚䈊㻲㼛㼡㼚㼐㼑㼐䈊

㻭㼡㼓㼡㼟㼠㻌㻝㻘㻌㻞㻜㻜㻜䈊㻯㼍㼜㼕㼠㼍㼘䈊

㻡㻘㻠㻡㻣㻚㻥㻌㼙㼕㼘㼘㼕㼛㼚㻌㼥㼑㼚㻌㻔㻲㼑㼎㼞㼡㼍㼞㼥㻌㻞㻜㻞㻟㻕䈊㻿㼠㼛㼏㼗㻌㻱㼤㼏㼔㼍㼚㼓㼑䈊㼀㼛㼗㼥㼛㻌㻿㼠㼛㼏㼗㻌㻱㼤㼏㼔㼍㼚㼓㼑㻌㻼㼞㼕㼙㼑㻌㻹㼍㼞㼗㼑㼠㻌㻔㻟㻢㻡㻢㻕䈊㻯㻱㻻䈊㻴㼕㼐㼑㼗㼍㼠㼟㼡㻌㻹㼛㼞㼕㼠㼍㻌㻔㻼㼞㼑㼟㼕㼐㼑㼚㼠㻌㼍㼚㼐㻌㻯㻱㻻㻕䈊㼅㼛㼟㼡㼗㼑㻌㻵㼓㼍㼞㼍㼟㼔㼕㻌㻔㼂㼕㼏㼑㻌㻯㼔㼍㼕㼞㼙㼍㼚㻕䈊㻴㼑㼍㼐㼝㼡㼍㼞㼠㼑㼞㼟㻌㻔㼀㼛㼗㼥㼛㻘㻌㻾㼛㼜㼜㼛㼚㼓㼕㻌㻴㼕㼘㼘㼟㻌㻹㼛㼞㼕㻌㼀㼛㼣㼑㼞㻕䈊㻻㼒㼒㼕㼏㼑㼟㻌㼕㼚㻌㻶㼍㼜㼍㼚䈊㻻㼟㼍㼗㼍㻌㻻㼒㼒㼕㼏㼑㻘㻌㻲㼡㼗㼡㼛㼗㼍㻌㻻㼒㼒㼕㼏㼑㻘㻌㻿㼑㼚㼐㼍㼕㻌㻻㼒㼒㼕㼏㼑䈊㻭㼒㼒㼕㼘㼕㼍㼠㼑㼐㻌㻯㼛㼙㼜㼍㼚㼕㼑㼟䈊㻳㼘㼛㼎㼍㼘㻌㻳㼑㼍㼞㻌㻵㼚㼏㻚䈊㻮㻸㻻㻯㻷㻿㻹㻵㼀㻴㻒㻯㼛㻚䈊㻵㼚㼠㼑㼞㼚㼍㼠㼕㼛㼚㼍㼘㻌㻻㼒㼒㼕㼏㼑䈊㻷㻸㼍㼎㻌㻯㼔㼕㼚㼍㻌㻵㼚㼏㻚䈊㻷㻸㼍㼎㻌㻳㼞㼛㼡㼜㻌㻱㼙㼜㼘㼛㼥㼑㼑㼟䈊㻡㻠㻝㻌㻔㻲㼡㼘㼘㻙㼠㼕㼙㼑㻌㼑㼙㼜㼘㼛㼥㼑㼑㼟㻌㼍㼟㻌㼛㼒㻌㻰㼑㼏㼑㼙㼎㼑㼞㻌㻞㻜㻞㻞㻕䈊4© KLab Inc.

㻭㼜㼜㻌㻿㼠㼛㼞㼑㻌㼀㼛㼜㻌㻿㼍㼘㼑㼟㻌㻾㼍㼚㼗㼕㼚㼓㼟䈊㼀㼛㼜㻌㻡㻜㻦䈊

㻤㻞䈊㻯㼛㼡㼚㼠㼞㼕㼑㼟䈊㻴㼛㼚㼓㻌㻷㼛㼚㼓 䈊㻹㼍㼏㼍㼛 䈊㼂㼕㼑㼠㼚㼍㼙 䈊㻲㼞㼍㼚㼏㼑 䈊㻻㼙㼍㼚 䈊
㻸㼑㼎㼍㼚㼛㼚 䈊㼁㻭㻱 䈊㼀㼡㼚㼕㼟㼕㼍 䈊㻼㼑㼞㼡 䈊

㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊

㻿㼕㼚㼓㼍㼜㼛㼞㼑 䈊㻹㼥㼍㼚㼙㼍㼞 䈊㻿㼜㼍㼕㼚 䈊㻿㼣㼕㼠㼦㼑㼞㼘㼍㼚㼐 䈊㻮㼍㼔㼞㼍㼕㼚 䈊㻷㼡㼣㼍㼕㼠 䈊㻱㼓㼥㼜㼠 䈊㻭㼞㼓㼑㼚㼠㼕㼚㼍 䈊㻱㼏㼡㼍㼐㼛㼞 䈊㼀㼛㼜㻌㻡㻜㻦䈊

㻢㻤㻌㻯㼛㼡㼚㼠㼞㼕㼑㼟䈊© KLab Inc.

㼀㼔㼍㼕㼘㼍㼚㼐䈊㻯㼍㼙㼎㼛㼐㼕㼍䈊㻵㼚㼐㼛㼚㼑㼟㼕㼍䈊㻹㼍㼘㼍㼥㼟㼕㼍䈊㻲㼕㼚㼘㼍㼚㼐䈊
㻹㼍㼏㼍㼛䈊
㻷㼛㼞㼑㼍䈊
㻮㼞㼡㼚㼑㼕䈊
㼀㼍㼕㼣㼍㼚䈊

㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊
㻝䈊

㼀㼛㼜㻌㻡㻜㻦䈊

㻣㻠㻌㻯㼛㼡㼚㼠㼞㼕㼑㼟䈊

㻝䈊
㻝䈊
㻟䈊
㻟䈊
㻟䈊
㻠䈊
㻠䈊
㻡䈊
㻢䈊

㻵㼞㼑㼘㼍㼚㼐䈊
㻴㼡㼚㼓㼍㼞㼥䈊
㻴㼛㼚㼓㻌㻷㼛㼚㼓䈊㻵㼠㼍㼘㼥䈊
㻺㼑㼣㻌㼆㼑㼍㼘㼍㼚㼐䈊㻼㼔㼕㼘㼘㼕㼜㼕㼚㼑㼟䈊㼂㼕㼑㼠㼚㼍㼙䈊
㻿㼕㼚㼓㼍㼜㼛㼞㼑䈊
㼁㻷䈊

㻣䈊
㻤䈊
㻥䈊
㻝㻜䈊
㻝㻝䈊
㻝㻝䈊
㻝㻝䈊
㻝㻟䈊
㻞㻟䈊

㻖㻰㼍㼠㼍㻌㼏㼛㼙㼜㼕㼘㼑㼐㻌㼎㼥㻌㻷㻸㼍㼎㻌㼡㼟㼕㼚㼓㻌㼞㼑㼟㼑㼍㼞㼏㼔㻌㼒㼞㼛㼙㻌㻭㼜㼜㻌㻭㼚㼚㼕㼑㻌㼟㼠㼍㼠㼕㼟㼠㼕㼏㼟㻌㼍㼢㼍㼕㼘㼍㼎㼘㼑㻌㼒㼞㼛㼙㻌㼑㼍㼏㼔㻌㼓㼍㼙㼑䇻㼟㻌㼞㼑㼘㼑㼍㼟㼑㻌㼐㼍㼠㼑㻌㼠㼛㻌㻿㼑㼜㼠㼑㼙㼎㼑㼞㻘㻌㻞㻜㻞㻝㻚䈊䈊㻮㼍㼔㼞㼍㼕㼚䈊
㻽㼍㼠㼍㼞䈊
㻸㼍㼠㼢㼕㼍䈊
㻵㼟㼞㼍㼑㼘䈊
㻷㼡㼣㼍㼕㼠䈊
㻳㼞㼑㼑㼏㼑䈊
㻲㼞㼍㼚㼏㼑䈊
㻮㼑㼘㼓㼕㼡㼙䈊
㻺㼑㼠㼔㼑㼞㼘㼍㼚㼐㼟䈊㼁㻭㻱䈊㻯㼍㼚㼍㼐㼍䈊
㻺㼛㼞㼣㼍㼥䈊
㻭㼡㼟㼠㼞㼕㼍䈊
㼁㻿㻭䈊
㼁㻷䈊

㻝䈊
㻝䈊
㻝䈊
㻟䈊
㻡䈊
㻡䈊
㻢䈊
㻢䈊
㻤䈊
㻤䈊
㻥䈊
㻥䈊
㻝㻜䈊
㻝㻥䈊
㻞㻠䈊

5

㻹㼛㼎㼕㼘㼑㻌㼓㼍㼙㼑㼟㻌㼍㼞㼑㻌㼍㻌㼓㼛㼛㼐㻌㼠㼑㼟㼠㻌㼎㼑㼐㻌㼒㼛㼞㻌㼙㼍㼠㼔㼑㼙㼍㼠㼕㼏㼍㼘㻌㼟㼏㼕㼑㼚㼏㼑䈊㻵㼐㼑㼍㼘㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚㼟㻌㼒㼛㼞㻌㼟㼏㼕㼑㼚㼏㼑䈊● 㻮㼕㼓㻌㼐㼍㼠㼍䈊○ 㻹㼕㼘㼘㼕㼛㼚㼟㻌㼛㼒㻌㼍㼏㼠㼕㼢㼑㻌㼡㼟㼑㼞㼟㻌䚓㻌㼀㻮㻙㼟㼏㼍㼘㼑㻌㼐㼍㼠㼍㻌㻛㻌㼐㼍㼥䈊○ 㻿㼑㼚㼟㼛㼞㼟㻌㼛㼚㻌㼜㼔㼛㼚㼑㼟㻦㻌㼠㼛㼡㼏㼔㻘㻌㼙㼕㼏㻘㻌㼏㼍㼙㼑㼞㼍㻘㻌㻳㻼㻿㻘㻌㼓㼥㼞㼛㻘㻌㻚㻚㻚䈊● 㻻㼜㼑㼚㻌㼜㼞㼛㼎㼘㼑㼙㼟䈊○ 㻭㼜㼜㼘㼕㼏㼍㼠㼕㼛㼚㼟㻦㻌㻳㼍㼙㼑㻌㻭㻵㻘㻌㻳㼍㼙㼑㻌㻰㼑㼢㻛㻻㼜㼟㻛㻭㼐㼟㻘㻌㻚㻚㻚䈊○ 㻯㼔㼍㼘㼘㼑㼚㼓㼑㼟㻦㻌㻾㼑㼍㼘㻙㼠㼕㼙㼑㻌㻹㻸㻌㼛㼚㻌㼑㼐㼓㼑㻌㼐㼑㼢㼕㼏㼑㼟㻘㻌㼞㼍㼜㼕㼐㻌㻹㻸㻻㼜㼟㻘㻌㻚㻚㻚䈊● 㻾㼍㼜㼕㼐㻌㼞㼑㼟㼑㼍㼞㼏㼔㻌㼏㼥㼏㼘㼑䈊○ 㻾㼑㼟㻌㼠㼑㼍㼙㻌㼐㼑㼜㼘㼛㼥㼟㻌㼑㼢㼑㼞㼥㻌㼣㼑㼑㼗㻌㼍㼚㼐㻌㼓㼑㼠㼟㻌㼕㼙㼙㼑㼐㼕㼍㼠㼑㻌㼒㼑㼑㼐㼎㼍㼏㼗㻌䈊© KLab Inc.

○ 㻺㼛㼠㻌㼙㼕㼟㼟㼕㼛㼚㻌㼏㼞㼕㼠㼕㼏㼍㼘㻦㻌㼣㼑㻌㼏㼍㼚㻌㼐㼛㻌㼠㼞㼕㼍㼘㼟㻙㼍㼚㼐㻙㼑㼞㼞㼛㼞㼟㻌㼕㼚㻌㼜㼞㼛㼐㼡㼏㼠㼟䈊169

6

㻹㼕㼟㼟㼕㼛㼚㻌㼛㼒㻌㻹㼍㼏㼔㼕㼚㼑㻌㻸㼑㼍㼞㼚㼕㼚㼓㻌㻳㼞㼛㼡㼜㻌㼕㼚㻌㻷㻸㼍㼎䈊㻾㼑㼟㼑㼍㼞㼏㼔㻌㼏㼡㼠㼠㼕㼚㼓㻙㼑㼐㼓㼑㻌㻹㻸㻌㼠㼑㼏㼔㼚㼛㼘㼛㼓㼕㼑㼟㻌㼠㼛㻌㼏㼞㼑㼍㼠㼑㻌㼡㼚㼜㼞㼑㼏㼑㼐㼑㼚㼠㼑㼐㻌㼓㼍㼙㼕㼚㼓㻌㼑㼤㼜㼑㼞㼕㼑㼚㼏㼑㼟䈊

Math

Data
Science

HPC

Game㻿㼕㼚㼓㼡㼘㼍㼞㼕㼠㼥㻌㼀㼔㼑㼛㼞㼥 䈊㻻㼜㼠㼕㼙㼕㼦㼍㼠㼕㼛㼚䈊㻹㼍㼏㼔㼕㼚㼑㻌㻸㼑㼍㼞㼚㼕㼚㼓 䈊㻰㼍㼠㼍㻌㻭㼚㼍㼘㼥㼟㼕㼟 䈊㻿㼡㼜㼑㼞㼏㼛㼙㼜㼡㼠㼑㼞 䈊㻯㼘㼛㼡㼐䈊

㻳㼍㼙㼑㻌㻰㼑㼢㻛㻻㼜㼟 䈊㻳㼍㼙㼑㻌㻭㻵䈊

㼣㼕㼠㼔㻌㻵㻹㻵㻘㻌㻷㼥㼡㼟㼔㼡㻌㼁㻚 䈊㼣㼕㼠㼔㻌㻾㻵㻷㻱㻺㻌㻭㻵㻼 䈊㼣㼕㼠㼔㻌㻾㻵㻵㼀㻘㻌㻷㼥㼡㼟㼔㼡㻌㼁㻚 䈊㼕㼚㻌㻷㻸㼍㼎䈊

7© KLab Inc.

㻭㼜㼜㼘㼕㼏㼍㼠㼕㼛㼚㼟㻌㼛㼒㻌㻹㻸㻌㼕㼚㻌㻷㻸㼍㼎䈊㻿㼜㼍㼙㻌㼒㼕㼘㼠㼑㼞㻌㼒㼛㼞㻌㼕㼚㻙㼓㼍㼙㼑㻌㼏㼔㼍㼠 䈊㻔㻳㼛㼛㼓㼘㼑㻌㻯㼘㼛㼡㼐㻌㻰㼍㼥㻦㻌㻰㼕㼓㼕㼠㼍㼘㻌䇻㻞㻞 㻕䈊㻯㼔㼍㼞㼠㻌㼓㼑㼚㼑㼞㼍㼠㼕㼛㼚㻌㼒㼛㼞㻌㼞㼔㼥㼠㼔㼙㻌㼓㼍㼙㼑㼟 䈊㻔㻯㻱㻰㻱㻯㻞㻜㻞㻝㻘㻌㻿㻵㻳㻳㻾㻭㻼㻴㻌㻭㼟㼕㼍㻌㻞㻜㻞㻝 㻘㻌㻭㻭㻭㻵㻙㻞㻟㻕䈊㻭㼡㼠㼛㼜㼘㼍㼥㻌㼒㼛㼞㻌㼁㻵㻌㼠㼑㼟㼠㼕㼚㼓 䈊㻔㻶㻿㻭㻵㻞㻜㻞㻞㻌㻭㼣㼍㼞㼐 㻕䈊㻟㻰㻌㼙㼛㼠㼕㼛㼚㻌㼞㼑㼠㼞㼕㼑㼢㼍㼘 䈊㻔㻵㻮㻵㻿㻞㻜㻞㻞㻕䈊㼍㼚㼐㻌㼙㼛㼞㼑䈊8© KLab Inc.

㻭㼞㼠㼕㼒㼕㼏㼕㼍㼘㻌㻳㼑㼚㼑㼞㼍㼘㻌㻵㼚㼠㼑㼘㼘㼕㼓㼑㼚㼏㼑㻌㻔㻭㻳㻵㻕㻌㼒㼛㼞㻌㻭㻾㻌㼓㼍㼙㼑㼟䈊
Touch / Gesture䈊㻭㻾㻌㼍㼓㼑㼚㼠㼟㻌㼟㼔㼛㼡㼘㼐㻌㼏㼛㼜㼑㻌㼣㼕㼠㼔䈊● 㼙㼡㼘㼠㼕㼙㼛㼐㼍㼘㻌㼕㼚㼜㼡㼠㼟㻛㼛㼡㼠㼜㼡㼠㼟䈊● 㼓㼍㼙㼑㻌㼟㼠㼍㼠㼑㼟䈊● 㼜㼘㼍㼚㼚㼕㼚㼓䈊㻯㼍㼙㼑㼞㼍䈊

㻳㻼㻿䈊㻾㼑㼍㼏㼠䈊

㻾㼑㼏㼛㼓㼚㼕㼦㼑䈊㻼㼘㼍㼚䈊㼂㼛㼕㼏㼑䈊

9© KLab Inc.

170

㻭㻾㻌㼍㼓㼑㼚㼠㻌㼍㼞㼏㼔㼕㼠㼑㼏㼠㼡㼞㼑䈊
User Input䈊

Recognition Routines䈊

Vector Embedding䈊

Text

Text Recognition

0.2

Audio
VisionGesture䠖

SpeechRecognition
ObjectRecognition
GestureRecognition

Real-World Info䈊

Virtual-World Info䈊GPS

NPC Position

Date / Time

NPC Emotion

WeatherGame State

Generative AI Routines䈊

1.4

Text Generation

Symbolic AI
Routines䈊䎩䈊
5.3

Planner䠖(KLab STRIPS)

2.1

Facial Expression
Character Animation
Emotional Speech
Sythesis

䎩䈊
4.6
䠖

NPC
Reaction

3.5

10© KLab Inc.

㻰㻺㻺㼟㻌㼕㼙㼜㼘㼑㼙㼑㼚㼠㻌㼞㼑㼏㼛㼓㼚㼕㼠㼕㼛㼚㻌㼍㼚㼐㻌㼞㼑㼍㼏㼠㼕㼛㼚㻌㻭㻵㼟䈊
User Input䈊

Recognition Routines

Text

Text Recognition

Audio
VisionGestureSpeechRecognition
ObjectRecognition
GestureRecognition

Real-World Info䈊

Game-World Info䈊GPS

NPC Position

Date / Time

NPC Emotion

WeatherGame State

Vector Embedding䈊
Text 0.2
generation +speech synthesis䠖Object detection +path finding

Navigation to tapped
position

Generative AI Routines䈊

1.4

Symbolic AI
Routines䈊䎩䈊
5.3

Planner䠖(KLab STRIPS)

2.1

Text Generation
Facial Expression
Character Animation
Emotional Speech
Sythesis

䎩䈊
4.6
䠖

NPC
Reaction

3.5

11© KLab Inc.

㻻㼜㼑㼚㻌㼜㼞㼛㼎㼘㼑㼙㻌㼕㼚㻌㼜㼘㼍㼚㼚㼕㼚㼓㻌㻭㻵䈊
User Input䈊

Text
Audio
VisionGestureHow do DNNs connect to symbolic AI routines?

Recognition Routines䈊

Vector Embedding䈊

Text Recognition

0.2
䠖

SpeechRecognition
ObjectRecognition
GestureRecognition

Real-World Info䈊

Virtual-World Info䈊GPS

NPC Position

Date / Time

NPC Emotion

WeatherGame State

Generative AI Routines䈊

1.4

䎩䈊

Symbolic AI
Routines䈊

5.3
䠖
2.1

䎩䈊

Planner(GPT STRIPS)

Text Generation
Facial Expression
Character Animation
Emotional Speech
Sythesis

4.6
䠖
3.5

NPC
Reaction
12© KLab Inc.

171

㻻㼡㼞㻌㼍㼜㼜㼞㼛㼍㼏㼔㻦㻌㻴㼡㼙㼍㼚㻙㻵㼚㻙㼀㼔㼑㻙㻺㼑㼡㼞㼛䈊Game Agent
Too many symbols!
We need to select truly
meaningful ones.
䋻 Use sparse modeling!

Human
Intervention
with XAIs

Recognizewith Recognition AIs
Symbols

Data

Planwith Inference AIs

Environment

Intention
React
with Generative AIs

Action13© KLab Inc.

㼀㼛㼜㼛㼘㼛㼓㼥㻘㻌㻹㼡㼘㼠㼕㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㻻㼜㼠㼕㼙㼕㼦㼍㼠㼕㼛㼚㻌㼍㼚㼐㻌㻿㼜㼍㼞㼟㼑㻌㻹㼛㼐㼑㼘㼕㼚㼓䈊

14© KLab Inc.

㻱㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼍㼚㼐㻌㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼜㼍㼠㼔䈊㻸㼕㼚㼑㼍㼞㻌㼞㼑㼓㼞㼑㼟㼟㼕㼛㼚㻦㻱㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻦㻻㻸㻿㻌㼑㼞㼞㼛㼞䈊㻔㼓㼛㼛㼐㼚㼑㼟㼟㻙㼛㼒㻙㼒㼕㼠㻕㻸㻝㻙㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻔㼟㼜㼍㼞㼟㼑㼚㼑㼟㼟㻕㻸㻞㻙㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻔㼞㼛㼎㼡㼟㼠㼚㼑㼟㼟㻕● 㻻㼜㼠㼕㼙㼕㼦㼑㻌㼣㼕㼠㼔㻌㼐㼕㼒㼒㼑㼞㼑㼚㼠㻌λ䇺㼟㻌㼍㼚㼐㻌㼒㼕㼤㼑㼐㻌α→ 㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼜㼍㼠㼔㻌㼒㼛㼞㻌α䈊●㼀㼛㻌㼠㼡㼚㼑㻌㼠㼣㼛㻌㼔㼥㼜㼑㼞㻙㼜㼍㼞㼍㼙㼟㻌(λ, α)㻘㻌㼣㼑㻌㼚㼑㼑㼐㻌㼠㼛㻌㼏㼛㼙㼜㼡㼠㼑㻌㼙㼍㼚㼥㻌㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼜㼍㼠㼔㼟㻌㼒㼛㼞㻌㼐㼕㼒㼒㼑㼞㼑㼚㼠㻌α’s䈊λ© KLab Inc.

172

15

㻲㼞㼛㼙㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼠㼛㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘㻌㼜㼞㼛㼎㼘㼑㼙䈊㻻㼡㼞㻌㼟㼠㼞㼍㼠㼑㼓㼥䈊㻝㻚 㻿㼔㼛㼣㻌㼍㼘㼘㻌㼙㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㼟㻌㼍㼞㼑㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘䈊㻞㻚 㻾㼑㼒㼛㼞㼙㼡㼘㼍㼠㼑㻌㼠㼔㼑㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼠㼛㻌㼍㻌㼙㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙䈊㻟㻚 㻱㼤㼠㼑㼚㼐㻌㼠㼔㼑㻌㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼜㼍㼠㼔㻌㼛㼚㻌㼠㼔㼍㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼍㼚㼐㻌㼍㼜㼜㼞㼛㼤㼕㼙㼍㼠㼑㻌㼕㼠㻌㼎㼥㻌㼍㻌㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤䈊16© KLab Inc.

㻹㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼛㼜㼠㼕㼙㼕㼦㼍㼠㼕㼛㼚㻌㼜㼞㼛㼎㼘㼑㼙㼟䈊㻻㼜㼠㼕㼙㼕㼦㼑㻌㼙㼡㼘㼠㼕㼜㼘㼑㻌㼒㼡㼚㼏㼠㼕㼛㼚㼟㻦䈊䈊
Pareto set䈊feasible set
gradients㻳㼛㼍㼘㻦㻌㼒㼕㼚㼐㻌㻼㼍㼞㼑㼠㼛㻌㼟㼑㼠㻌㻔㼞㼍㼠㼔㼑㼞㻌㼠㼔㼍㼚㻌㼍㻌㼜㼛㼕㼚㼠㻕䈊㻰㼑㼒㼕㼚㼕㼠㼕㼛㼚㻦㻌㻭㻌㼜㼛㼕㼚㼠㻌x㻌㼕㼟㻌㼍㻌㻼㼍㼞㼑㼠㼛㻌㼟㼛㼘㼡㼠㼕㼛㼚㻌㼕㼒㻌㼠㼔㼑㼞㼑㻌㼕㼟㻌㼚㼛㻌y㻌㼟㼡㼏㼔㻌㼠㼔㼍㼠䈊
Pareto front

17© KLab Inc.

㼃㼑㼍㼗㼘㼥㻌㻿㼕㼙㼜㼘㼕㼏㼕㼍㼘㻌㻼㼞㼛㼎㼘㼑㼙㼟㻌㼇㻷㼛㼎㼍㼥㼍㼟㼔㼕㻗㻌㻞㻜㻝㻥㼉䈊condition 1

condition 2

Cr-surjective
Cr-embedding

weakly simplicial

Cr-weakly simplicial if two

Cr-surjective map
a Cr-embedding

where an (M - 1)-simplex isan I-subsimplex isfor each18© KLab Inc.

173㻭㻌㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤㻌㼍㼜㼜㼞㼛㼤㼕㼙㼍㼠㼑㼟㻌㼠㼔㼑㻌㻼㼍㼞㼑㼠㼛㻌㼟㼑㼠㻌㼛㼒㻌㼍㼚㼥㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘㻌㼜㼞㼛㼎㼘㼑㼙 䈊

A Bezier simplex is apolynomial map of
w1
degree D:

w3
b(w)
w2

Simplex ΔM-1

Hyper-surface in RK

19© KLab Inc.

㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤㻌㼒㼕㼠㼠㼕㼚㼓㻌㼇㼀㼍㼚㼍㼗㼍㻗㻌㻞㻜㻞㻜㼉䈊

㻭㻭㻻䈊

㻵㻿㻷䈊

© KLab Inc.

㼇㼀㼍㼚㼍㼗㼍㻗㻌㻞㻜㻞㻜㼉㻌㻭㼟㼥㼙㼜㼠㼛㼠㼕㼏㻌㻾㼕㼟㼗㻌㼛㼒㻌㻮㽴㼦㼕㼑㼞㻌㻲㼕㼠㼠㼕㼚㼓䈊

20㻮㼑㼦㼕㼑㼞㻌㻲㼘㼛㼣㻦㻌㻭㻌㻿㼡㼞㼒㼍㼏㼑㻙㼣㼕㼟㼑㻌㻳㼞㼍㼐㼕㼑㼚㼠㻌㻰㼑㼟㼏㼑㼚㼠㻌㻹㼑㼠㼔㼛㼐㻌㼇㻿㼍㼚㼚㼍㼕㻗㻌㻞㻜㻞㻞㼉䈊© KLab Inc.

㼇㻿㼍㼚㼚㼍㼕㻗㻌㻞㻜㻞㻞㼉㻌㻮㽴㼦㼕㼑㼞㻌㻲㼘㼛㼣㻦㻌㼍㻌㻿㼡㼞㼒㼍㼏㼑㻙㼣㼕㼟㼑㻌㻳㼞㼍㼐㼕㼑㼚㼠㻌㻰㼑㼟㼏㼑㼚㼠㻌㻹㼑㼠㼔㼛㼐㻌㼒㼛㼞㻌㻹㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㻻㼜㼠㼕㼙㼕㼦㼍㼠㼕㼛㼚䈊

174

21

㻿㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼙㼍㼜㼜㼕㼚㼓㼟䈊22© KLab Inc.

㻼㼞㼑㼢㼕㼛㼡㼟㻌㼣㼛㼞㼗䈊㻼㼞㼑㼢㼕㼛㼡㼟㻌㼣㼛㼞㼗㻌㼟㼔㼛㼣㼟㼇㻴㼍㼙㼍㼐㼍㻗㻌㻞㻜㻞㻜㼉㼇㻴㼍㼙㼍㼐㼍㻗㻌㻞㻜㻞㻜㼉㼇㻴㼍㼙㼍㼐㼍㻗㻌㻞㻜㻞㻝㼉㼇㻴㼍㼙㼍㼐㼍㻗㻌㻞㻜㻞㻜㼉㻌㻺㻚㻌㻴㼍㼙㼍㼐㼍㻘㻌㻷㻚㻌㻴㼍㼥㼍㼚㼛㻘㻌㻿㻚㻌㻵㼏㼔㼕㼗㼕㻘㻌㼅㻚㻌㻷㼍㼎㼍㼠㼍㻘㻌㼍㼚㼐㻌㻴㻚㻌㼀㼑㼞㼍㼙㼛㼠㼛㻘㻌㼀㼛㼜㼛㼘㼛㼓㼥㻌㼛㼒㻌㻼㼍㼞㼑㼠㼛㻌㼟㼑㼠㼟㻌㼛㼒㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㼟㻘㻌㻿㻵㻭㻹㻌㻶㻚㻌㻻㼜㼠㼕㼙㻚㻌㻟㻜㻘㻌㼚㼛㻚㻌㻟㻘㻌㻞㻢㻡㻥䇵㻞㻢㻤㻢㻚㼇㻴㼍㼙㼍㼐㼍㻗㻌㻞㻜㻞㻝㼉㻌 㻺㻚㻌㻴㼍㼙㼍㼐㼍㻌㼍㼚㼐㻌㻿㻚㻌㻵㼏㼔㼕㼗㼕㻘㻌㻿㼕㼙㼜㼘㼕㼏㼕㼍㼘㼕㼠㼥㻌㼛㼒㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㼟㻘㻌㻶㻚㻌㻹㼍㼠㼔㻚㻌㻿㼛㼏㻚㻌㻶㼍㼜㼍㼚㻚㻌㻣㻟㻘㻌㼚㼛㻚㻌㻟㻘㻌㻥㻢㻡䇵㻥㻤㻞㻚

䈊

䈊

23

© KLab Inc.

㻻㼡㼠㼘㼕㼚㼑㻌㼛㼒㻌㼜㼞㼛㼛㼒䈊㼀㼔㼡㼟㻘㻌㼠㼔㼑㻌㼜㼞㼛㼎㼘㼑㼙㻌㼛㼒㻌㼙㼕㼚㼕㼙㼕㼦㼕㼚㼓㻌㼒㻌㼕㼟㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘㻚24© KLab Inc.

175

㻲㼞㼛㼙㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼠㼛㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘㻌㼜㼞㼛㼎㼘㼑㼙䈊㻻㼡㼞㻌㼟㼠㼞㼍㼠㼑㼓㼥䈊㻝㻚 㻿㼔㼛㼣㻌㼍㼘㼘㻌㼙㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㼟㻌㼍㼞㼑㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘䈊㻞㻚 㻾㼑㼒㼛㼞㼙㼡㼘㼍㼠㼑㻌㼠㼔㼑㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼠㼛㻌㼍㻌㼙㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙䈊㻟㻚 㻱㼤㼠㼑㼚㼐㻌㼠㼔㼑㻌㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼜㼍㼠㼔㻌㼛㼚㻌㼠㼔㼍㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼍㼚㼐㻌㼍㼜㼜㼞㼛㼤㼕㼙㼍㼠㼑㻌㼕㼠㻌㼎㼥㻌㼍㻌㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤䈊25© KLab Inc.

㻹㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠䈊㻿㼕㼚㼓㼘㼑㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㻦㻌㼏㼛㼚㼢㼑㼤

㻦㻌㼏㼛㼚㼢㼑㼤

㻦㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻹㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㼏㼛㼚㼢㼑㼤

㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤26© KLab Inc.

㻲㼞㼛㼙㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼠㼛㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘㻌㼜㼞㼛㼎㼘㼑㼙䈊㻻㼡㼞㻌㼟㼠㼞㼍㼠㼑㼓㼥䈊㻝㻚 㻿㼔㼛㼣㻌㼍㼘㼘㻌㼙㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙㼟㻌㼍㼞㼑㻌㼣㼑㼍㼗㼘㼥㻌㼟㼕㼙㼜㼘㼕㼏㼕㼍㼘䈊㻞㻚 㻾㼑㼒㼛㼞㼙㼡㼘㼍㼠㼑㻌㼠㼔㼑㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼠㼛㻌㼍㻌㼙㼡㼘㼠㼕㻙㼛㼎㼖㼑㼏㼠㼕㼢㼑㻌㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼜㼞㼛㼎㼘㼑㼙䈊㻟㻚 㻱㼤㼠㼑㼚㼐㻌㼠㼔㼑㻌㼞㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㼜㼍㼠㼔㻌㼛㼚㻌㼠㼔㼍㼠㻌㼜㼞㼛㼎㼘㼑㼙㻌㼍㼚㼐㻌㼍㼜㼜㼞㼛㼤㼕㼙㼍㼠㼑㻌㼕㼠㻌㼎㼥㻌㼍㻌㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤䈊27© KLab Inc.

176

㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻹㼍㼜㻦㻌㻞㻙㻼㼍㼞㼍㼙㼑㼠㼑㼞㻌㻱㼤㼠㼑㼚㼟㼕㼛㼚㻌㼛㼒㻌㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻼㼍㼠㼔䈊(α, λ) 䌜 [0, ∞)² → (w1, w2, w3) 䌜 Δ2

28

© KLab Inc.

㻱㼤㼜㼑㼞㼕㼙㼑㼚㼠㼟䈊㻝㻚㻲㼛㼞㻌㼑㼍㼏㼔㻌㼐㼍㼠㼍㼟㼑㼠㻘㻌㼠㼞㼍㼕㼚㻌㻡㻘㻝㻡㻝㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㼟㻌㼣㼕㼠㼔㻌㼐㼕㼒㼒㼑㼞㼑㼚㼠㻌㼣㼑㼕㼓㼔㼠㼟㻌㼛㼚㻌䂴 㻞䈊㻞㻚㻿㼜㼘㼕㼠㻌㼍㻌㼟㼑㼠㻌㼛㼒㻌㼠㼞㼍㼕㼚㼑㼐㻌㼑㼘㼍㼟㼠㼕㼏㻌㼚㼑㼠㼟㻌㼕㼚㼠㼛㻌㼠㼞㼍㼕㼚㼕㼚㼓㻛㼠㼑㼟㼠㻌㼟㼑㼠㼟㻌㼒㼛㼞㻌㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤㻌㼒㼕㼠㼠㼕㼚㼓 䈊㻟㻚㼀㼞㼍㼕㼚㻌㼍㻌㻮㼑㼦㼕㼑㼞㻌㼟㼕㼙㼜㼘㼑㼤㻌㼣㼕㼠㼔㻌㼠㼞㼍㼕㼚㼕㼚㼓㻌㼟㼑㼠㻌㼍㼚㼐㻌㼑㼢㼍㼘㼡㼍㼠㼑㻌㼕㼠㼟㻌㼑㼞㼞㼛㼞㻌㼣㼕㼠㼔㻌㼠㼑㼟㼠㻌㼟㼑㼠 䈊29© KLab Inc.

㻭㻌㻮㼑㼦㼕㼑㼞㻌㻿㼕㼙㼜㼘㼑㼤㻌㻯㼛㼚㼢㼑㼞㼓㼑㼟㻌㼛㼚㻌㼠㼔㼑㻌㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻹㼍㼜㻌㼍㼟㻌㼚㻌䊻㻌䌲䈊30© KLab Inc.

177

㼀㼔㼑㻌㻻㼜㼠㼕㼙㼍㼘㻌㻰㼑㼓㼞㼑㼑㻌㼛㼒㻌㼍㻌㻮㼑㼦㼕㼑㼞㻌㻿㼕㼙㼜㼘㼑㼤㻌㼕㼟㻌㻺㼛㼠㻌㻿㼑㼚㼟㼕㼠㼕㼢㼑㻌㼠㼛㻌㻰㼍㼠㼍㼟㼑㼠㼟 䈊

31© KLab Inc.

㻾㼑㼓㼡㼘㼍㼞㼕㼦㼍㼠㼕㼛㼚㻌㻹㼍㼜㻦㻌㻳㼞㼛㼡㼚㼐㻌㼀㼞㼡㼠㼔㻌㼢㼟㻚㻌㻭㼜㼜㼞㼛㼤㼕㼙㼍㼠㼕㼛㼚䈊32© KLab Inc.

㻳㼑㼚㼑㼞㼍㼘㼕㼦㼍㼠㼕㼛㼚䈊㼍㼚㼐㻌㻼㼛㼕㼚㼏㼍㼞㽴㻌㻯㼛㼚㼖㼑㼏㼠㼡㼞㼑䈊33© KLab Inc.

178

㻳㼑㼚㼑㼞㼍㼘㼕㼦㼍㼠㼕㼛㼚㻌㼠㼛㻌㼚㼛㼚㻙㼟㼠㼞㼛㼚㼓㼘㼥㻌㼏㼛㼚㼢㼑㼤㻌㼏㼍㼟㼑㼟㻦㻌㻿㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㻌㼠㼑㼟㼠㻌㼛㼒㻌㼟㼕㼙㼜㼘㼕㼏㼕㼠㼥 䈊
We focus onchecking it

We skip it since it
is easy to checkcondition 1

condition 2

Cr-surjectiver

C -embedding● 㼃㼔㼑㼚㻌㼠㼔㼑㻌㼟㼕㼙㼜㼘㼕㼏㼕㼠㼥㻌㼛㼒㻌㼠㼔㼑㻌㼜㼞㼛㼎㼘㼑㼙㻌㼕㼟㻌㼡㼚㼗㼚㼛㼣㼚㻘㻌㼣㼑㻌㼚㼑㼑㼐㻌㼠㼛㻌㼑㼟㼠㼕㼙㼍㼠㼑㻌㼕㼠㻌㼒㼞㼛㼙㻌㼐㼍㼠㼍䈊● 㼃㼑㻌㼟㼠㼍㼠㼕㼟㼠㼕㼏㼍㼘㼘㼥㻌㼠㼑㼟㼠㻌㻯㼛㼚㼐㼕㼠㼕㼛㼚㻌㻝㻌㼒㼞㼛㼙㻌㼍㻌㼟㼍㼙㼜㼘㼑㻌㼛㼒㻌㻼㼍㼞㼑㼠㼛㻌㼟㼑㼠㻌㼍㼚㼐㻌㼒㼞㼛㼚㼠䈊34© KLab Inc.

㻹㼍㼗㼑㻌㼠㼔㼑㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚㻌㻝㻌㼑㼍㼟㼕㼑㼞㻦㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚㻌㻝㻌䋽㻌㼏㼛㼚㼐㼕㼠㼕㼛㼚㼟㻌㻝㼍㻌㻒㻌㻝㼎䈊

We focus onchecking it

We skip it since it
is easy to check

Cond 1a
Cond 1b35© KLab Inc.

㻴㼛㼙㼛㼘㼛㼓㼥㻌㻱㼟㼠㼕㼙㼍㼠㼕㼛㼚㻌㼁㼟㼕㼚㼓㻌㻯㼛㼚㼒㼕㼐㼑㼚㼏㼑㻌㻿㼑㼠㻌㼛㼒㻌㻼㼑㼞㼟㼕㼟㼠㼑㼚㼠㻌㻰㼕㼍㼓㼞㼍㼙 䈊

Stability theorem of persistence diagrams:
α-confidence intervalof noise36© KLab Inc.

179

㼀㼔㼑㼞㼑㻌㼕㼟㻌㼍㻌㼙㼕㼟㼟㼕㼚㼓㻌㼘㼕㼚㼗㻚㻚㻚䈊㻵㼚㻌㼓㼑㼚㼑㼞㼍㼘㻘䈊

Topological spaces
X and Yare homeomorphic

Homologies
H*(X) and H*(Y)are isomorphic

We will seek a condition
where the converse is true.
37© KLab Inc.

㻵㼚㻌㼍㻌㼏㼑㼞㼠㼍㼕㼚㻌㼍㼟㼟㼡㼙㼜㼠㼕㼛㼚㻘㻌㼔㼛㼙㼛㼘㼛㼓㼥㻌㼕㼙㼜㼘㼕㼑㼟㻌㼔㼛㼙㼑㼛㼙㼛㼞㼜㼔㼕㼟㼙䈊䋽 n-dimensional Poincaré conjecture38© KLab Inc.

㻼㼞㼛㼛㼒䈊

39© KLab Inc.

180

㼀㼍㼗㼑㼍㼣㼍㼥㼟䈊㻹㼛㼎㼕㼘㼑㻌㼓㼍㼙㼑㼟㻌㼎㼞㼕㼐㼓㼑㻌㼎㼑㼠㼣㼑㼑㼚㻌㼙㼍㼠㼔㻌㼍㼚㼐㻌㼕㼚㼐㼡㼟㼠㼞㼥䈊

Math

Data
Science

HPC

Game㻿㼕㼚㼓㼡㼘㼍㼞㼕㼠㼥㻌㼀㼔㼑㼛㼞㼥 䈊㻻㼜㼠㼕㼙㼕㼦㼍㼠㼕㼛㼚䈊㻹㼍㼏㼔㼕㼚㼑㻌㻸㼑㼍㼞㼚㼕㼚㼓 䈊㻰㼍㼠㼍㻌㻭㼚㼍㼘㼥㼟㼕㼟 䈊㻿㼡㼜㼑㼞㼏㼛㼙㼜㼡㼠㼑㼞 䈊㻯㼘㼛㼡㼐䈊

㻳㼍㼙㼑㻌㻰㼑㼢㻛㻻㼜㼟 䈊㻳㼍㼙㼑㻌㻭㻵䈊

㼣㼕㼠㼔㻌㻵㻹㻵㻘㻌㻷㼥㼡㼟㼔㼡㻌㼁㻚 䈊㼣㼕㼠㼔㻌㻾㻵㻷㻱㻺㻌㻭㻵㻼 䈊㼣㼕㼠㼔㻌㻾㻵㻵㼀㻘㻌㻷㼥㼡㼟㼔㼡㻌㼁㻚 䈊㼕㼚㻌㻷㻸㼍㼎䈊

㼃㼑㻌㼍㼘㼣㼍㼥㼟㻌㼣㼑㼘㼏㼛㼙㼑㻌㼏㼛㼘㼘㼍㼎㼛㼞㼍㼠㼕㼛㼚㻌㼣㼕㼠㼔㻌㼥㼛㼡㻍䈊40© KLab Inc.

Copyright (c) KLab Inc. All Rights Reserved.䈊

41

181

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Persistent Homology and Machine Learning
Yuichi Ike
Institute of Mathematics for Industry, Kyushu University, Japan
Persistent homology is a central tool in topological data analysis. It encodes the topological features of given data into persistence diagrams, which are multisets in thetwo-dimensional space. In connection with machine learning, persistence diagrams
have been used as an input of machine learning algorithms as feature vectors and are
effectively applied in material science and medical science. Recently, many techniques
have been developed to incorporate persistence diagrams into loss functions for controlling the topology of parameters. In this talk, I will start with the basics of persistenthomology and some applications. Then I would like to discuss several recent developments in optimizing TDA-based loss functions and their applications in dimensionality
reduction or visualization.

183

Persistent Homology and Machine Learning
WORKSHOP on Mathematics for Industry
2023-09-27

Yuichi Ike
Institute of Mathematics for Industry, Kyushu University
Joint work with
Mathieu Carrière, Frédéric Chazal, Marc Glisse,
Hariprasad Kannan, Théo Lacombe, Martin Royer, Yuhei Umeda(Collaboration with Inria and Fujitsu)

1/26

Outline1. Persistent Homology and Applications• “Shape” of data and persistence diagrams• Typical applications of persistent homology

2. PH-based Loss Functions• Differentiability of persistence diagrams• Applications of PH-based loss functions

Optimize

2/26

Persistent Homology and Applications• Extracting the shape of data
• Persistent homology and persistence diagrams (PDs)• Some applications

3/26

184

Idea of persistent homology (PH)
Topological Data Analysis (TDA)◼Method to extract topological features of data

One hole

Without hole

Q. How to extract the “topology” of a discrete point cloud?

4/26

Idea of persistent homology (PH)
Topological Data Analysis (TDA)◼Method to extract topological features of data

One hole

Without hole

Q. How to extract the “topology” of a discrete point cloud?
◼Idea1: Consider the union of balls centered at data points

5/26

Idea of persistent homology (PH)
Topological Data Analysis (TDA)◼Method to extract topological features of data

One hole

Without hole

Q. How to extract the “topology” of a discrete point cloud?
◼Idea1: Consider the union of balls centered at data points

6/26

185

Idea of persistent homology (PH)
Topological Data Analysis (TDA)◼Method to extract topological features of data

One hole

Without hole

Q. How to extract the “topology” of a discrete point cloud?
◼Idea1: Consider the union of balls centered at data points-> How to adjust the radius value of balls?

7/26

Idea of persistent homology (PH)
Topological Data Analysis (TDA)◼Method to extract topological features of data

One hole

Without hole

Q. How to extract the “topology” of a discrete point cloud?
◼Idea1: Consider the union of balls centered at data points-> How to adjust the radius value of balls?

◼Idea2: Consider all radii and track the evolution: persistent homology-> Can distinguish noise and essential topological features

8/26

Filtrations on simplicial complexes
Filtration: increasing family of subcomplexes◼Simplicial complex: collection 𝐾𝐾 ⊂ 2𝑉𝑉 s.t. 𝜎𝜎 ∈ 𝐾𝐾, 𝜏𝜏 ⊂ 𝜎𝜎 ⇒ 𝜏𝜏 ∈ 𝐾𝐾
◼𝒦𝒦 = 𝐾𝐾𝑟𝑟 𝑟𝑟 , 𝐾𝐾𝑟𝑟 ⊂ 𝐾𝐾 filtration of 𝐾𝐾: ⇔ 𝐾𝐾𝑟𝑟 ⊂ 𝐾𝐾𝑠𝑠 (𝑟𝑟 ≤ 𝑠𝑠) and ‫𝐾𝐾 = 𝑟𝑟𝐾𝐾 𝑟𝑟ڂ‬↔ Function 𝑓𝑓: 𝐾𝐾 → ℝ s.t. 𝜎𝜎 ⊂ 𝜏𝜏 ⇒ 𝑓𝑓 𝜎𝜎 ≤ 𝑓𝑓(𝜏𝜏), 𝐾𝐾𝑟𝑟 = {𝜎𝜎 ∈ 𝐾𝐾 ∣ 𝑓𝑓 𝜎𝜎 ≤ 𝑟𝑟}
2

0

1

0
0

1

0

0

Čech filtration

2

1

𝑥𝑥0 , … 𝑥𝑥𝑘𝑘 ∈ 𝐶𝐶(𝑃𝑃; 𝑟𝑟) :⇔ ‫∅ ≠ 𝑟𝑟 ; 𝑖𝑖𝑥𝑥 𝐵𝐵 𝑖𝑖ځ‬Rips filtration

𝑥𝑥0 , … 𝑥𝑥𝑘𝑘 ∈ 𝑅𝑅 𝑃𝑃; 𝑟𝑟 :⇔ 𝐵𝐵 𝑥𝑥𝑖𝑖 ; 𝑟𝑟 ∩ 𝐵𝐵 𝑥𝑥𝑗𝑗 ; 𝑟𝑟 ≠ ∅ ∀𝑖𝑖, 𝑗𝑗
9/26

186

PH and persistence diagrams (PDs)
Filtration: increasing family of subcomplexes◼Simplicial complex: collection 𝐾𝐾 ⊂ 2𝑉𝑉 s.t. 𝜎𝜎 ∈ 𝐾𝐾, 𝜏𝜏 ⊂ 𝜎𝜎 ⇒ 𝜏𝜏 ∈ 𝐾𝐾
◼𝒦𝒦 = 𝐾𝐾𝑟𝑟 𝑟𝑟 , 𝐾𝐾𝑟𝑟 ⊂ 𝐾𝐾 filtration of 𝐾𝐾: ⇔ 𝐾𝐾𝑟𝑟 ⊂ 𝐾𝐾𝑠𝑠 (𝑟𝑟 ≤ 𝑠𝑠) and ‫𝐾𝐾 = 𝑟𝑟𝐾𝐾 𝑟𝑟ڂ‬↔ Function 𝑓𝑓: 𝐾𝐾 → ℝ s.t. 𝜎𝜎 ⊂ 𝜏𝜏 ⇒ 𝑓𝑓 𝜎𝜎 ≤ 𝑓𝑓(𝜏𝜏), 𝐾𝐾𝑟𝑟 = {𝜎𝜎 ∈ 𝐾𝐾 ∣ 𝑓𝑓 𝜎𝜎 ≤ 𝑟𝑟}

1

Death time

2

1

0
0

∞

2

0

1
0

0

2

1

Conn. comp.
Loops

Birth time1

Persistent homology of 𝒦𝒦 = 𝐾𝐾𝑟𝑟 𝑟𝑟 is the family⋯ → 𝐻𝐻𝑛𝑛 𝐾𝐾𝑟𝑟 → 𝐻𝐻𝑛𝑛 𝐾𝐾𝑠𝑠 → 𝐻𝐻𝑛𝑛 𝐾𝐾𝑡𝑡 → ⋯ 𝑟𝑟 ≤ 𝑠𝑠 ≤ 𝑡𝑡
⇝Persistence diagram (PD): encodes the birth and death time ofeach homology class
10/26

How to use PDs◼Points far from the diagonal express essential shapes while thosenear the diagonal are regarded as noises◼We can analyze which type of shape is represented by a point in PD

Connected
components

Small loop

Loops

Big loop

11/26

How to use PDs

More directly ...
We can distinguish data points with different “shapes”
No holeConnected
components

One hole
Loops

12/26

187

Analysis with persistent homology
Data

Filtrations

MachineLearningPDsĞĐƚŽƌŝǌĂƚŝŽŶ
Example• Point clouds generated
by simulation• Graphs• Image datae.g., persistence image, persistence landscape

13/26

Analysis of silica glass

Kusano, Fukumizu, and Hiraoka: Persistence weighted Gaussian
kernel for topological data analysis, ICML2016◼cf. Nakamura et al.: Description of medium-range order in amorphous
structures by persistent homology.
◼Estimate the temperature that SiO₂ changes from liquid to glass state◼Idea: Transform point clouds into PDs and analyze them>ŝƋƵŝĚ
PD+'ůĂƐƐ

Vectorization of PDs(They used kernel method)

Figures from K. Fukumizu, Persistence Weighted
Gaussian Kernel for Topological Data Analysis

14/26

Application to graph classification
PH extracts some global structure of graphs,
which can be used for classification◼Need to find suitable filtrations on graphse.g., degree function, Heat Kernel Signature

PersLay, M. Carrière, F. Chazal, I., T. Lacombe,
M. Royer, Y. Umeda, AISTATS 2020

◼Proposed a new architecture for graph classification◼PersLay (NN vectorization) + one-layer NN

More and more studies to combine PH
and Machine Learning
15/26

188

PH-based Loss Functions• Applications of PH-based loss functions• Differentiability and convergence of PH-based functions• Applications

16/26

PH-based loss functions◼Many attempts to construct PH-based loss functionsℒ: loss function

data◼Brüel-Gabrielsson et al., A Topology Layer for Machine Learning,
AISTATS2020: deformation of point clouds, topological generative models◼Moor et al., Topological Autoencoders, ICML2020: Topology-preserving AE

17/26

Parametrized filtrations and PDs
Recall: A filtration of a simplicial complex 𝐾𝐾↔ a vector 𝑓𝑓 ∈ ℝ𝐾𝐾 s.t. 𝜎𝜎 ⊂ 𝜏𝜏 ⇒ 𝑓𝑓𝜎𝜎 ≤ 𝑓𝑓𝜏𝜏Filt 𝐾𝐾 ≔ 𝑓𝑓 ∈ ℝ𝐾𝐾 𝜎𝜎 ⊂ 𝜏𝜏 ⇒ 𝑓𝑓𝜎𝜎 ≤ 𝑓𝑓𝜏𝜏 }A parametrized filtration: a function 𝐹𝐹: 𝐴𝐴 → Filt 𝐾𝐾 , where 𝐴𝐴 ⊂ ℝ𝑑𝑑◼Rips filtration
𝐹𝐹: (ℝ𝑑𝑑 )𝑁𝑁 → ℝ|Δ𝑁𝑁 | , 𝐹𝐹𝜎𝜎 𝑥𝑥 ≔ 1Τ2 max𝑖𝑖,𝑗𝑗∈𝜎𝜎 ‖𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑗𝑗 ‖◼Parameters in ML 𝑓𝑓𝜃𝜃 : ℝ𝑑𝑑 → ℝ𝐷𝐷 𝜃𝜃 ∈ Θ , 𝑃𝑃 ⊂ ℝ𝑑𝑑 : finite subset
𝐹𝐹: Θ → ℝ|Δ𝑁𝑁 | , 𝐹𝐹𝜎𝜎 𝜃𝜃 ≔ 1Τ2 max𝑖𝑖,𝑗𝑗∈𝜎𝜎 𝑓𝑓𝜃𝜃 𝑥𝑥𝑖𝑖 − 𝑓𝑓𝜃𝜃 𝑥𝑥𝑗𝑗

The PD is a vector in ℝ|𝐾𝐾| : 𝑝𝑝1 , … , 𝑝𝑝𝑚𝑚 , 𝑒𝑒1 , … , 𝑒𝑒𝑛𝑛 , 𝑝𝑝𝑖𝑖 ∈ ℝ2 , 𝑒𝑒𝑖𝑖 ∈ ℝ⇒ The assignment is viewed as 𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏: 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅 𝑲𝑲 → ℝ|𝑲𝑲| persistence map
𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏 ∘ 𝑭𝑭: 𝑨𝑨 → ℝ|𝑲𝑲| parametrized PD

189

18/26

Optimization of PH-based functions◼Q. How can we optimize PH-based loss functions?
◼A. Usually just apply gradient descent◼For a differentiable function ℒ: 𝐴𝐴 → ℝ, 𝐴𝐴 ⊂ ℝ𝑑𝑑 , update the parameter by
𝑥𝑥𝑘𝑘+1 = 𝑥𝑥𝑘𝑘 − 𝛼𝛼𝑘𝑘 ∇ℒ 𝑥𝑥𝑘𝑘 ,where 𝛼𝛼𝑘𝑘 is the learning rate at step 𝑘𝑘◼Toy example: optimize a point set to maximize “# of loops”◼For the 1st PD 𝐷𝐷1 (𝑃𝑃), considerℒ 𝑃𝑃 = − ෍𝑝𝑝∈𝐷𝐷1 𝑃𝑃

𝑝𝑝 − 𝜋𝜋Δ (𝑝𝑝) 2∞ + 𝑑𝑑 𝑃𝑃, 𝐶𝐶 ,where 𝜋𝜋Δ is the projection to the diagonal and 𝐶𝐶 is the square
Optimize with 𝓛𝓛

19/26

Differentiability of persistence map◼How to compute PDs from filtrations?

Death time

2

0
1

0
1

0

Loops

2

0

0

Birth time

2

1

1

1. Find the pairs of birth and death simplices

𝜎𝜎𝑏𝑏𝑖𝑖 , 𝜎𝜎𝑑𝑑𝑖𝑖

2. Associate the filtration value to each pair𝐹𝐹 𝜎𝜎𝑏𝑏𝑖𝑖 , 𝐹𝐹 𝜎𝜎𝑑𝑑𝑖𝑖

e.g.

: birth: deathe.g. (1,2)◼𝐹𝐹 is smoothly parametrized => can consider 𝑥𝑥 ↦𝑖𝑖

(combinatorial)𝑖𝑖∇𝑥𝑥 𝐹𝐹 𝜎𝜎𝑏𝑏𝑖𝑖 , ∇𝑥𝑥 𝐹𝐹 𝜎𝜎𝑑𝑑𝑖𝑖

in the area where the order of simplices does not change

𝑖𝑖

20/26

PH-based functions and subdifferential
Function of PDs: a permutation invariant function 𝐸𝐸: ℝ|𝐾𝐾| → ℝ
𝐸𝐸 𝑝𝑝𝛼𝛼 1 , … , 𝑝𝑝𝛼𝛼 𝑚𝑚 , 𝑒𝑒𝛽𝛽 1 , … , 𝑒𝑒𝛽𝛽 𝑛𝑛 = 𝐸𝐸(𝑝𝑝1 , … , 𝑝𝑝𝑚𝑚 , 𝑒𝑒1 , … , 𝑒𝑒𝑛𝑛 )e.g.

◼Distance between PDs,
◼Persistence landscape,◼Persistence image, …

If 𝐹𝐹 and 𝐸𝐸 are in a good class, then so is ℒ = 𝐸𝐸 ∘ Pers ∘ 𝐹𝐹: 𝐴𝐴 → ℝ and
it is differentiable a.e.
=> can define the subdifferential
𝜕𝜕𝜕(𝑧𝑧) = Conv lim ∇ℒ 𝑧𝑧𝑖𝑖 : ℒ is differentiable at 𝑧𝑧𝑖𝑖
𝑧𝑧𝑖𝑖 →𝑧𝑧

Figures from GUDHI Library webpage

21/26

190

Convergence of PH-based functions
We can apply stochastic (sub)gradient descent to optimize
PH-based loss functions using automatic differentiationℒ: loss function

data◼However, there was no guarantee of convergence

Carrière, Chazal, Glisse, I., Kannan, and Umeda,
Optimizing persistent homology based functions, ICML2021◼Proved the almost surely convergence of stochastic subgradientdescent for a wide class of PH-based loss functions◼The class includes almost all the PH-based functions in the literature

22/26

Convergence result◼Theorem 𝐾𝐾 simplicial compex, 𝐹𝐹: 𝐴𝐴 → ℝ|𝐾𝐾| parametrized family of filtration,
𝐸𝐸: ℝ|𝐾𝐾| → ℝ function of PDs, ℒ = 𝐸𝐸 ∘ Pers ∘ 𝐹𝐹: 𝐴𝐴 → ℝ
Assume that 𝑭𝑭 and 𝑬𝑬 are in a good class (definable) and 𝓛𝓛 is locally Lipschitz.
Consider the sequence obtained by𝑦𝑦𝑘𝑘 ∈ 𝜕𝜕ℒ 𝑥𝑥𝑘𝑘 ,𝑥𝑥𝑘𝑘+1 = 𝑥𝑥𝑘𝑘 − 𝛼𝛼𝑘𝑘 𝑦𝑦𝑘𝑘 + 𝜉𝜉𝑘𝑘 ,where 𝛼𝛼𝑘𝑘 : learning rate and 𝜉𝜉𝑘𝑘 : random variable s.t.
ͳǤ 𝛼𝛼𝑘𝑘 ≥ 0, σ𝑘𝑘 𝛼𝛼𝑘𝑘 = ∞, σ𝑘𝑘 𝛼𝛼𝑘𝑘2 < ∞ ;ʹǤ sup 𝑥𝑥𝑘𝑘 < ∞ almost surely;𝑘𝑘

3. For ℱ𝑘𝑘 = 𝜎𝜎(𝑥𝑥𝑗𝑗 , 𝑦𝑦𝑗𝑗 , 𝜉𝜉𝑗𝑗 , 𝑗𝑗 < 𝑘𝑘) , there exists a function 𝑝𝑝: ℝ𝑑𝑑 → ℝ that isbounded on any bounded set s.t. for any 𝑘𝑘 almost surely
𝔼𝔼 𝜉𝜉𝑘𝑘 2 ℱ𝑘𝑘 < 𝑝𝑝 𝑥𝑥𝑘𝑘 .
𝔼𝔼 𝜉𝜉𝑘𝑘 ℱ𝑘𝑘 = 0,
Then 𝒙𝒙𝒌𝒌 𝒌𝒌 converges to a critical point of 𝓛𝓛 almost surely.

cf. Davis et al., Stochastic subgradient method converges on tame functions, 2020

23/26

Applications: Filtration learning
Learn filtrations to give PDs for a classification task

◼Consider the toy task to classify the MNIST images with 0th PDs + RF
◼For a linear function 𝑓𝑓 to some direction, consider
σ𝑦𝑦𝑖𝑖=𝑦𝑦𝑗𝑗 =𝑙𝑙 𝑑𝑑 𝐷𝐷0 𝐼𝐼𝑖𝑖 , 𝑓𝑓 , 𝐷𝐷0 𝐼𝐼𝑗𝑗 , 𝑓𝑓.
ℒ 𝑓𝑓 = ෍σ𝑦𝑦𝑖𝑖=𝑙𝑙 𝑑𝑑 𝐷𝐷0 𝐼𝐼𝑖𝑖 ; 𝑓𝑓 , 𝐷𝐷0 𝐼𝐼𝑗𝑗 ; 𝑓𝑓𝑙𝑙Optimize 𝓛𝓛 𝒇𝒇 to find the best direction.

Classification accuracy before and after filter optimization

191

24/26

Applications: Graph Filtration Learning
Hofer et al., Graph Filtration Learning, ICML2020◼Learn a filter function of graphs end-to-end◼Recall that a function on vertices gives a filtration of a graph
2

1

3

1

4
1

⊂

2

⊂

3

⊂

◼Vectorization of the resulting PDs is used for classification

4◼Parametrized vertex filter function can be implemented by GNN andlearned thanks to the differentiability and the convergence result
25/26

Summary

1. Persistent Homology and Applications⚫Extract “topology” of data as persistence diagrams (PDs)⚫We can use PD as input of machine learning (ML)⚫Applications: material science, graph classification, …

2. PH-based Loss Functions⚫Many attempts to combine PH and ML⚫Proved the convergence of SSGD for a PH-based loss functions⚫Developing thanks to differentiability of PDs
Optimize⚫ Filtration Leaning, Topologically Regularized Embeddings, …

Thank you for your attention!

192

26/26

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Exotic shapes of nano-spherical structures - new
DNA coding
Stanislaw Janeczko
Center for Advanced Studies,
Warsaw University of Technology, Poland(joint work with Hassan Babiker)
The simplest naturally ordered tetrahedral packing consists of an ordered sequence ofregular tetrahedra glued together face to face as with the linear packing of a tetrahedral
helix.Such tetrahedral structures are called tetrahedral chains.
Any tetrahedral chain consists of the three types of simplest configurations of four
consecutive tetrahedra called tetrahedral units. Two of these types are left and righttetrahedral short spirals, U, D, and the third type, F, is a flat configuration of four
tetrahedra. The structure of a tetrahedral chain in D, F, U elementary units is written
as a word like U U DF U D....
The three strands of the left or right oriented tetrahedral helix form a spiral withirrational slope. This is the reason for the effective density of tetrahedral chains andnonexistence of closed tetrahedral chains in Euclidean space.
Let us assume that the gluing process of tetrahedra is ordered along a chain and eachstep of this process is realized by reflection in a particular face of adjacent tetrahedron.
To each tetrahedron we assign four reflections Ri , i = 1, . . . , 4, in the configurational
three dimensional space V. Reflections Ri in V are represented by four correspondingreflect-morphisms R̄i , i = 1, . . . , 4, acting in the space of regular tetrahedra T througha reflectional transformation of their vertices. In V, dim V = n, any tetrahedral chainof length n + 1 is uniquely represented by an initial tetrahedron T and an orderedsequence of n reflect-morphisms
R̄i1 , . . . , R̄in ,ik ̸= ik+1 , k = 1, . . . , n − 1.

The fact that a tetrahedral chain is so rigid in 3-space and regular tetrahedra can not
tile the space gives rise to several questions. The main question we consider is therecognition of combinatorial and algebraic structures of tetrahedral chains. We want
to investigate their geometric properties and determine what kind of information iscontained in the chain invariants of orthogonal transformations and re-numberings. We
use the parametrization of the chains by sequences of ordered reflections in barycentric
coordinates and find their combinatorial structure. Periodicity along a chain is based on
the structure of sequences of admissible triplets of integers and their cycling properties.
The corresponding numerical invariants and an indexing role of a binary tetrahedralgroup defines the complete coding properties in dimension three.

193

Exotic shapes of nano-sphericalstructures - new DNA coding

Stanislaw Janeczko
Faculty of Mathematics and Information Sciences, Warsaw University of Technology

Workshop on Mathematics for Industry
Kyushu - Warsaw 25-29 September 2023
CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

1

Sphere packings- Square packing, face-centered cubic packing

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

2- Barrow boy’s packing, cell is a rhombic dodecahedron

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

194

3

- Hexagonal packing, the third layer sits exactly above
the first layer.

CAS-MINI Warsaw 25 - 29 September 2023,

Stanis­law Janeczko

4

Stanislaw Janeczko

5

Stanislaw Janeczko

6

Sphere packing

CAS-MINI Warsaw 25 - 29 September 2023,

Sphere packing

CAS-MINI Warsaw 25 - 29 September 2023,

195

Sphere packing

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

7

Stanislaw Janeczko

8

Icosahedron

CAS-MINI Warsaw 25 - 29 September 2023,

Tetrahedral chains

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

196

9

Tetrahedral chains

H. Steinhaus, 1957; J.H. Mason, 1972
CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

10

Stanislaw Janeczko

11

Stanislaw Janeczko

12

Tetrahedral chains

CAS-MINI Warsaw 25 - 29 September 2023,

Tetrahedral chains

CAS-MINI Warsaw 25 - 29 September 2023,

197

Almost closed tetrahedral chains

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

13

Stanislaw Janeczko

14

Dual tetrahedral chains

CAS-MINI Warsaw 25 - 29 September 2023,

Tetrahedra in barycentric coordinatesT ≡ {p1, p2, p3, p4}, {(S1, p1), . . . , (S4, p4)}
T −regular tetrahedra,  pi − pj = pk − pl , i = j, k = l
T ⊂ V ⊗ U ∗,U ≡ R4V - configurational affine space, dimV = 3
U - barycentric coordinates (α1, . . . , α4) ∈ U

H={ 4i=1 αi = 1} - canonical affine hyperplaneCAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

198

15


∗
T ∈ T ,T = 4
i=1 pi ⊗ ei

Barycentric coordinate map T : H → V :4∗T(α) = 4i=1 pi ⊗ ei (α) = i=1 αipi,

α= 4
i=1 αiei ∈ H, and geometrically

T = T(H ∩ {αi ≥ 0})

F : V → V affine mapping.

F lifts to a linear mappingM : (U, H) → (U, H)
preserving the hyperplane H
CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

16

M is defined uniquely by the commuting diagram
T(M (•)) = F (T(•))
F (pi) = 4
j=1 αjipj in barycentric coordinates αji.

Then
4


4


i=1 j=1

αjipj ⊗e∗i =

4


j=1

pj ⊗(

4


i=1

αjie∗i ) =CAS-MINI Warsaw 25 - 29 September 2023,

4


j=1

pj ⊗M ∗(e∗j ).

Stanislaw Janeczko

17

Generation of tetrahedral chain
4
si center of Si, si = 1
3 ( j=1 pj − pi)Four orthogonal reflections by Si
Ri(p) = p − 2
Ri(pj ) = pj + 2δij ((p − si|si − pi)(si − pi)(si − pi|si − pi)
1 
pk − pj ),3 k=ij = 1, . . . , 4{T (i)}ni=0 tetrahedral chain

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

199

18

T (0) = T,(1)

= R̄i1 T,(2)
Ti1i2 = R̄i2 R̄i1 T,
T i1

...

i1 = i2,

...

...

(n)
Ti1i2...in = R̄in . . . R̄i2 R̄i1 T,

ik+1 = ik , k = 1, . . . , n − 1.

R̄i : T → T twist morphisms, defined by Ri.
CAS-MINI Warsaw 25 - 29 September 2023,

Stanis­law Janeczko

19

Representation in barycentric coordinatesR̄i(v ⊗ u∗) = v ⊗ Mi∗u∗R̄i : T → T ,


M1 =













M3 =



T





T



T





T

2 2 2
0 0 0
−1 3
 1
3 3




 2
2 2


−1
0 1 0 0


 3
3
3
 , M2 = 
 ,




 0
0
1
0
0 0 1 0






0 0 0 10 0 0 10 
1 0 0 0
 1 0 0
 0 1 00 
0 1 0 0




 .
2 −1 2 
 , M4 =  0 0 1

0 


3
3

 2 2 2

−1
0 0 0 1
3 3 3







 2

 3


CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

20

R̄i is represented by transpose of Mi
EXAMPLE

R̄1(

4


i=1

pi ⊗ e∗i ) =

4


i=1

(1)1

⊗ e∗i ,





pi

where














(1)1

p1
p
p






(1)1 

2(1)1 


3

(1) 

p4



=

1












2 2
−1 2
 p 
3 3 3
 1 


0 1 0 0  p2 

,


 p 
0 0 1 0 3 

p4
0 0 0 1(n)Ti1...in = R̄in . . . R̄i1 T.
CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

200

21

Coding in triplets of consecutive steps(r+1)= R̄k T (r)
(r+2)
Tkj
= R̄j R̄k T (r)
(r+3)
Tkji
= R̄iR̄j R̄k T (r).
Tk

(r+3)

= R̄iR̄j R̄k T (r)U, D, F :

Tkji

F :

T (r+3);det(xr+1, xr+2, xr+3) = 0

U :T (r+3);det(xr+1, xr+2, xr+3) > 0

D:T (r+3);det(xr+1, xr+2, xr+3) < 0

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

23

Stanislaw Janeczko

24

Shape orientation

c(3)
x3

c

(2)

x2

c (1)
x1

c(0)

CAS-MINI Warsaw 25 - 29 September 2023,

Basic units

Tetrahedral chains: DDU F . . . U DF F D.

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

201

25

Combinatorial codes for U, D, F
Admissible triplets parametrizing U D F:
(k, i, j), 1 ≤ i, j, k ≤ 4,k = j = i

EXAMPLEU U DF D(3, 4, 2) → (4, 2, 1) → (2, 1, 4) → (1, 4, 1) → (4, 1, 3).
(7)

T3421413 = R3R1R4R1R2R4R3T

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

26

UUUUUUUUUUFFUUUUUUUUUUFUFUUUUUUUUUUFFUUUUUUUUUUFF

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

27

UUUUUUUUUUFFUUUUUUUUUUFUFUUUUUUUUUUFFUUUUUUUUUUFF

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

202

28

Classification of admissible tripletsu√det(x1, x2, x3) = 32 3/243(k, j, i)(3, 2, 1)(4, 3, 1)(2, 4, 1)(4, 1, 2)(1, 3, 2)(3, 4, 2)(2, 1, 3)(4, 2, 3)(1, 4, 3)(3, 1, 4)(1, 2, 4)(2, 3, 4)d√det(x1, x2, x3) = −32 3/243(k, j, i)(4, 2, 1)(2, 3, 1)(3, 4, 1)(3, 1, 2)(4, 3, 2)(1, 4, 2)(4, 1, 3)(1, 2, 3)(2, 4, 3)(2, 1, 4)(3, 2, 4)(1, 3, 4)

CAS-MINI Warsaw 25 - 29 September 2023,fdet(x1, x2, x3) = 0(k, j, i)(1, 2, 1)(1, 3, 1)(1, 4, 1)(2, 1, 2)(2, 3, 2)(2, 4, 2)(3, 1, 3)(3, 2, 3)(3, 4, 3)(4, 1, 4)(4, 2, 4)(4, 3, 4)

Stanislaw Janeczko

29

U -chains period(3, 2, 1) → (2, 1, 4) → (1, 4, 3) → (4, 3, 2)(4, 3, 1) → (3, 1, 2) → (1, 2, 4) → (2, 4, 3)(2, 4, 1) → (4, 1, 3) → (1, 3, 2) → (3, 2, 4)(3, 4, 2) → (4, 2, 1) → (2, 1, 3) → (1, 3, 4)(4, 1, 2) → (1, 2, 3) → (2, 3, 4) → (3, 4, 1)(4, 2, 3) → (2, 3, 1) → (3, 1, 4) → (1, 4, 2)(1, 4, 3) → (4, 3, 2) → (3, 2, 1) → (2, 1, 4)(1, 2, 4) → (2, 4, 3) → (4, 3, 1) → (3, 1, 2)(1, 3, 2) → (3, 2, 4) → (2, 4, 1) → (4, 1, 3)(2, 1, 3) → (1, 3, 4) → (3, 4, 2) → (4, 2, 1)(2, 3, 4) → (3, 4, 1) → (4, 1, 2) → (1, 2, 3)(3, 1, 4) → (1, 4, 2) → (4, 2, 3) → (2, 3, 1)

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

30

D-chains period(2, 1, 4) → (1, 4, 3) → (4, 3, 2) → (3, 2, 1)(3, 1, 2) → (1, 2, 4) → (2, 4, 3) → (4, 3, 1)(4, 1, 3) → (1, 3, 2) → (3, 2, 4) → (2, 4, 1)(4, 2, 1) → (2, 1, 3) → (1, 3, 4) → (3, 4, 2)(1, 2, 3) → (2, 3, 4) → (3, 4, 1) → (4, 1, 2)(2, 3, 1) → (3, 1, 4) → (1, 4, 2) → (4, 2, 3)(4, 3, 2) → (3, 2, 1) → (2, 1, 4) → (1, 4, 3)(2, 4, 3) → (4, 3, 1) → (3, 1, 2) → (1, 2, 4)(3, 2, 4) → (2, 4, 1) → (4, 1, 3) → (1, 3, 2)(1, 3, 4) → (3, 4, 2) → (4, 2, 1) → (2, 1, 3)(3, 4, 1) → (4, 1, 2) → (1, 2, 3) → (2, 3, 4)(1, 4, 2) → (4, 2, 3) → (2, 3, 1) → (3, 1, 4)

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

203

31

Combinatorial structure
I = {(α, β) ∈ ∆ × ∆ : α = β}
∆ = {1, 2, 3, 4}
Uniquely defined mappings
Lu, Ld, Lf : I → ∆,#I = 12and bijections
Lu, Ld, Lf : I → I,
L∗(i1, i2) = (i2, L∗(i1, i2)), ∗ = u, d, f.
CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

32

L - sequence for tetrahedral chain
ExampleDU U F D−→

L d Lf Lu Ld Ld

Any periodic tetrahedral chain is characterized bycycling composition of a numerical representation of
its period

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

33

Compositions of L∗−sequences form the indexing
space for tetrahedral chains

The indexing space is a binary tetrahedral subgroupof S12
generated by three elements Lu,

relationsL3u = id,L3
d = id,L2
f = id,CAS-MINI Warsaw 25 - 29 September 2023,

Ld ,Lf with the(LuLd)2 = id.

Stanislaw Janeczko

204

35

Geometric characteristics-proper tetrahedral chainsn 3 4 5 6 7
8
9
10
11
12
13
An 1 3 9 26 76 218 628 1802 5146 14670 41734-branching order 0 ≤ b ≤ 3-vertex order P (p),-clustering functionp∈VCn P (p) = 4n

Cl(Cn) =p∈VCn

max(0, P (p) − 4)

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

36

Stanislaw Janeczko

37

Stanislaw Janeczko

38

Tetrahelix

CAS-MINI Warsaw 25 - 29 September 2023,

Zero branching order

CAS-MINI Warsaw 25 - 29 September 2023,

205

Proper chains sharing one common vertex
b\n
1
2
3
total

4
0
0
2
2

5
0
0
4
4

6
0
1
6
7

7
0
4
9
13

8
0
6
16
22

9
1
10
27
38

10
2
24
38
64

11
6
46
48
100

12
9
78
55
142

CAS-MINI Warsaw 25 - 29 September 2023,

13
19
113
56
188

14
38
137
50
225

15
49
153
35
237

16
69
132
22
223

17
79
85
12
176

18
71
36
2
109

Stanis­law Janeczko

19
34
6
0
40

20
6
0
0
6

total
383
831
380
1594

39

Ico-clusters
F F U F F DU DU DF F U F F DU, F F U F F DU DU DU DF F U F F
U F F DF F U DU DU F F DF F U, U F F DU DF F U F F DU DF F UU DF F U F F DU DF F U F F DU, U DF F U F F DU DU DF F U F F.

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

40

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

41

206

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

42

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

43

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

44

Smallest unit b = 1

207

Clustering folding

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

45

Stanislaw Janeczko

46

Stanislaw Janeczko

47

Clustering folding

CAS-MINI Warsaw 25 - 29 September 2023,

Big periodic

CAS-MINI Warsaw 25 - 29 September 2023,

208

Mixed clustering folding

H. Babiker, S. Janeczko, Combinatorial representation of tetrahedralchains, Communications in Information and Sciences, Vol. 15, No. 3, (2015),
331-359

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

48

Stanislaw Janeczko

49

Stanislaw Janeczko

50

Nano-blood particles

CAS-MINI Warsaw 25 - 29 September 2023,

Spray technology

CAS-MINI Warsaw 25 - 29 September 2023,

209

Silica particles

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

51

Stanislaw Janeczko

52

Stanislaw Janeczko

53

Large silica particles

CAS-MINI Warsaw 25 - 29 September 2023,

Porous particles

CAS-MINI Warsaw 25 - 29 September 2023,

210

Stable porous particles

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

54

Stanislaw Janeczko

55

Porous particles

CAS-MINI Warsaw 25 - 29 September 2023,

Hollow particles

S.Y. Lee, L. Gradon, S. Janeczko, F. Iskandar, K. Okuyama, Formationof Highly Ordered Nanostructures by drying Micrometer Colloidal
Droplets, ACS Nano Journal, Vol. 4, No. 8, (2010), 4717-4724
L. Gradon, S. Janeczko, M. Abdullah, F. Iskandar, K. Okuyama, SelfOrganization Kinetics of Mesoporous Nanostructured Particles, AIChE
Journal Vol. 50, No. 10, (2004), 2583-2593.
CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

211

56

Geometrical coding of DNA sequencesI{ FFUFFDUDUDUDFFUFF,{ A,C} } ACACTCTACTACTATACAC{ FFUFFDUDUDUDFFUFF,{ A,T} } ATATGTGATGATGAGATAT{ FFUFFDUDUDUDFFUFF,{ A,G} } AGAGCGCAGCAGCACAGAG{ FFUFFDUDUDUDFFUFF,{ C,A} } CACAGAGCAGCAGCGCACA{ FFUFFDUDUDUDFFUFF,{ C,T} } CTCTATACTACTACACTCT{ FFUFFDUDUDUDFFUFF,{ C,G} } CGCGTGTCGTCGTCTCGCG{ FFUFFDUDUDUDFFUFF,{ T,A} } TATACACTACTACTCTATA{ FFUFFDUDUDUDFFUFF,{ T,C} } TCTCGCGTCGTCGTGTCTC{ FFUFFDUDUDUDFFUFF,{ T,G} } TGTGAGATGATGATATGTG{ FFUFFDUDUDUDFFUFF,{ G,A} } GAGATATGATGATGTGAGA{ FFUFFDUDUDUDFFUFF,{ G,C} } GCGCACAGCAGCAGAGCGC{ FFUFFDUDUDUDFFUFF,{ G,T} } GTGTCTCGTCGTCGCGTGTII{ FFUFFDUDUDFFUFFDU,{ A,C} } ACACTCTACTACACTCTAC{ FFUFFDUDUDFFUFFDU,{ A,T} } ATATGTGATGATATGTGAT{ FFUFFDUDUDFFUFFDU,{ A,G} } AGAGCGCAGCAGAGCGCAG{ FFUFFDUDUDFFUFFDU,{ C,A} } CACAGAGCAGCACAGAGCA{ FFUFFDUDUDFFUFFDU,{ C,T} } CTCTATACTACTCTATACT{ FFUFFDUDUDFFUFFDU,{ C,G} } CGCGTGTCGTCGCGTGTCG{ FFUFFDUDUDFFUFFDU,{ T,A} } TATACACTACTATACACTA{ FFUFFDUDUDFFUFFDU,{ T,C} } TCTCGCGTCGTCTCGCGTC{ FFUFFDUDUDFFUFFDU,{ T,G} } TGTGAGATGATGTGAGATG{ FFUFFDUDUDFFUFFDU,{ G,A} } GAGATATGATGAGATATGA{ FFUFFDUDUDFFUFFDU,{ G,C} } GCGCACAGCAGCGCACAGC{ FFUFFDUDUDFFUFFDU,{ G,T} } GTGTCTCGTCGTGTCTCGT
III{ UDFFUFFDUDUDFFUFF,{ A,C} } ACTATACACTACTATACAC{ UDFFUFFDUDUDFFUFF,{ A,T} } ATGAGATATGATGAGATAT{ UDFFUFFDUDUDFFUFF,{ A,G} } AGCACAGAGCAGCACAGAG{ UDFFUFFDUDUDFFUFF,{ C,A} } CAGCGCACAGCAGCGCACA{ UDFFUFFDUDUDFFUFF,{ C,T} } CTACACTCTACTACACTCT{ UDFFUFFDUDUDFFUFF,{ C,G} } CGTCTCGCGTCGTCTCGCG{ UDFFUFFDUDUDFFUFF,{ T,A} } TACTCTATACTACTCTATA{ UDFFUFFDUDUDFFUFF,{ T,C} } TCGTGTCTCGTCGTGTCTC{ UDFFUFFDUDUDFFUFF,{ T,G} } TGATATGTGATGATATGTG{ UDFFUFFDUDUDFFUFF,{ G,A} } GATGTGAGATGATGTGAGA{ UDFFUFFDUDUDFFUFF,{ G,C} } GCAGAGCGCAGCAGAGCGC{ UDFFUFFDUDUDFFUFF,{ G,T} } GTCGCGTGTCGTCGCGTGT
1

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

57

Stanislaw Janeczko

58

Geometrical coding of DNA sequencesIV{ UDFFUFFDUDFFUFFDU,{ A,C} } ACTATACACTACACTCTAC{ UDFFUFFDUDFFUFFDU,{ A,T} } ATGAGATATGATATGTGAT{ UDFFUFFDUDFFUFFDU,{ A,G} } AGCACAGAGCAGAGCGCAG{ UDFFUFFDUDFFUFFDU,{ C,A} } CAGCGCACAGCACAGAGCA{ UDFFUFFDUDFFUFFDU,{ C,T} } CTACACTCTACTCTATACT{ UDFFUFFDUDFFUFFDU,{ C,G} } CGTCTCGCGTCGCGTGTCG{ UDFFUFFDUDFFUFFDU,{ T,A} } TACTCTATACTATACACTA{ UDFFUFFDUDFFUFFDU,{ T,C} } TCGTGTCTCGTCTCGCGTC{ UDFFUFFDUDFFUFFDU,{ T,G} } TGATATGTGATGTGAGATG{ UDFFUFFDUDFFUFFDU,{ G,A} } GATGTGAGATGAGATATGA{ UDFFUFFDUDFFUFFDU,{ G,C} } GCAGAGCGCAGCGCACAGC{ UDFFUFFDUDFFUFFDU,{ G,T} } GTCGCGTGTCGTGTCTCGTV{ UFFDUDFFUFFDUDFFU,{ A,C} } ACTCTACTCTATACTATAC{ UFFDUDFFUFFDUDFFU,{ A,T} } ATGTGATGTGAGATGAGAT{ UFFDUDFFUFFDUDFFU,{ A,G} } AGCGCAGCGCACAGCACAG{ UFFDUDFFUFFDUDFFU,{ C,A} } CAGAGCAGAGCGCAGCGCA{ UFFDUDFFUFFDUDFFU,{ C,T} } CTATACTATACACTACACT{ UFFDUDFFUFFDUDFFU,{ C,G} } CGTGTCGTGTCTCGTCTCG{ UFFDUDFFUFFDUDFFU,{ T,A} } TACACTACACTCTACTCTA{ UFFDUDFFUFFDUDFFU,{ T,C} } TCGCGTCGCGTGTCGTGTC{ UFFDUDFFUFFDUDFFU,{ T,G} } TGAGATGAGATATGATATG{ UFFDUDFFUFFDUDFFU,{ G,A} } GATATGATATGTGATGTGA{ UFFDUDFFUFFDUDFFU,{ G,C} } GCACAGCACAGAGCAGAGC{ UFFDUDFFUFFDUDFFU,{ G,T} } GTCTCGTCTCGCGTCGCGT
VI{ UFFDFFUDUDUFFDFFU,{ A,C} } ACTCTATACTACTCTATAC{ UFFDFFUDUDUFFDFFU,{ A,T} } ATGTGAGATGATGTGAGAT{ UFFDFFUDUDUFFDFFU,{ A,G} } AGCGCACAGCAGCGCACAG{ UFFDFFUDUDUFFDFFU,{ C,A} } CAGAGCGCAGCAGAGCGCA{ UFFDFFUDUDUFFDFFU,{ C,T} } CTATACACTACTATACACT{ UFFDFFUDUDUFFDFFU,{ C,G} } CGTGTCTCGTCGTGTCTCG{ UFFDFFUDUDUFFDFFU,{ T,A} } TACACTCTACTACACTCTA{ UFFDFFUDUDUFFDFFU,{ T,C} } TCGCGTGTCGTCGCGTGTC{ UFFDFFUDUDUFFDFFU,{ T,G} } TGAGATATGATGAGATATG{ UFFDFFUDUDUFFDFFU,{ G,A} } GATATGTGATGATATGTGA{ UFFDFFUDUDUFFDFFU,{ G,C} } GCACAGAGCAGCACAGAGC{ UFFDFFUDUDUFFDFFU,{ G,T} } GTCTCGCGTCGTCTCGCGT
1

CAS-MINI Warsaw 25 - 29 September 2023,

A = Adenine, T = Thymine, C = Cytosine, G = Guanine

CAS-MINI Warsaw 25 - 29 September 2023,

Stanislaw Janeczko

212

59

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

How to measure data diversity and why it isimportant?
Pawel Józiak
Faculty of Mathematics and Computer Science,Warsaw University of Technology, Poland
In Machine Learning we often hear about patterns that algorithms overfit to. To
prevent it, a high quality data, a bunch of data that is curated needs to be prepared.
I will discuss what tools are available, other than manual labor, in order to tell whether
the dataset is diverse, and how we used the knowledge gained through it in order to prepare a highly diverse (and thus highly challenging) Document Understanding Dataset
and Evaluation (DUDE) in the domain of DocumentAI, a field at the boundary of
Natural Language Processing and Computer Vision. Joint work with Jordy Van Landeghem, Rubén Tito, L ukasz Borchmann, Michal Pietruszka, Rafal Powalski, Dawid
Jurkiewicz, Mickaël Coustaty, Bertrand Ackaert, Ernest Valveny, Matthew Blaschko,
Sien Moens, Tomasz Stanislawek.
References[1] Jordy Van Landeghem, Rubén Tito, L ukasz Borchmann, Michal Pietruszka, Rafal Powalski,
Dawid Jurkiewicz, Mickaël Coustaty, Bertrand Ackaert, Ernest Valveny, Matthew Blaschko,
Sien Moens, Tomasz Stanislawek. Document Understanding Dataset and Evaluation (DUDE).
Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2023, pp.
19528-19540

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Cryptographic protocol verification - results of
EPW project
Konstanty Junosza-Szaniawski
Faculty of Mathematics and Information Science,Warsaw University of Technology, Poland
Cryptographic protocols are fundamental to cybersecurity, necessitating assurance thatthese protocols are devoid of flaws. Among the various tools available for the verification of cryptographic protocols, ProVerif stands out. ProVerif models protocols
using Horn formulas and verifies the security properties through the satisfiability ofcorresponding logical formulas. However, the complexity of modeling protocols andtheir properties in ProVerif is time-consuming and requires a high level of knowledge.
To address this, we have developed a translator from the AnB language, which describes protocols from a global perspective, to ProVerif syntax. This translator simplifies the modeling process, enabling easy verification of key security properties with
ProVerif, such as secrecy, forward secrecy, weak secrecy, indistinguishability, authentication, non-replay authentication, and key compromise impersonation. Our translator
is a principal outcome of the project ”Experimental Platform for Automatic Validation of Crypto Algorithms and Verification of Crypto Protocols” (EPW), funded by
The National Centre for Research and Development under the grant CYBERSECIDENT/456962/III/NCBR/2020.

229

Cryptographic protocol verification
results of EPW projectElżbieta Andrukiewicz, Daniel Waszkiewicz
National Institute of Telecommunications, Poland
Tomasz Brengos, Anna Cichocka, Konstanty Junosza-Szaniawski, Adam Komorowski,
Agata Pilitowska, Hubert Grochowski,
Warsaw University of Technology, Poland

Supported by The National Centre for Research and Development
CYBERSECIDENT/456962/III/NCBR/2020

R+D Project EPW –brief presentationǆƉĞƌŝŵĞŶƚĂůƉůĂƚĨŽƌŵĨŽƌĂƵƚŽŵĂƚŝĐǀĂůŝĚĂƚŝŽŶŽĨĐƌǇƉƚŽĂůŐŽƌŝƚŚŵƐ ĂŶĚǀĞƌŝĨŝĐĂƚŝŽŶŽĨĐƌǇƉƚŽƉƌŽƚŽĐŽůƐ;ĂĐƌŽŶǇŵ͗WtͿ⮚WƌŽũĞĐƚĨŝŶĂŶĐĞĚƵŶĚĞƌƚŚĞŶĂƚŝŽŶĂůƉƌŽŐƌĂŵƐƉŽŶƐŽƌĞĚďǇEĂƚŝŽŶĂůĞŶƚƌĞŽĨ
Research and Development „Cybersecurity and eͲIdentity” (ǇďĞƌ^ĞĐ/ĚĞŶƚͿ⮚dŚĞŽŶƐŽƌƚŝƵŵĐŽŶƐŝƐƚƐŽĨϯZΘWŽůŝƐŚ ĞŶƚŝƚŝĞƐ͗▪ EĂƚŝŽŶĂů/ŶƐƚŝƚƵƚĞŽĨdĞůĞĐŽŵŵƵŶŝĐĂƚŝŽŶƐ– ^ƚĂƚĞZĞƐĞĂƌĐŚ/ŶƐƚŝƚƵƚĞ;ŽŶƐŽƌƚŝƵŵ>ĞĂĚĞƌͿ▪ E^<– ^ƚĂƚĞZĞƐĞĂƌĐŚ/ŶƐƚŝƚƵƚĞ
▪ tĂƌƐĂǁhŶŝǀĞƌƐŝƚǇŽĨdĞĐŚŶŽůŽŐǇ͕DĂƚŚĞŵĂƚŝĐƐĂŶĚ/ŶĨŽƌŵĂƚŝŽŶ ^ĐŝĞŶĐĞ&ĂĐƵůƚǇ⮚ƚŝŵĞĨƌĂŵĞ͗:ƵůǇϮϬϮϬ– ĞĐĞŵďĞƌϮϬϮϯ

EPW original concept
Lab
A

Lab
B

Lab
C
Dockers containing ref.
implementationsǆƉĞƌŝŵĞŶƚĂůsĂůŝĚĂƚŝŽŶWůĂƚĨŽƌŵ;WtͿ
Implementatios underverification

Allows self-assessment!
VendorM

VendorN

230

Formal verification of cryptoalgorithms

Equivalence checking (1)Equivalence checking’ (EC) for two implementationsŵĞĂŶƐƚŚĂƚƚǁŽŝŵƉůĞŵĞŶƚĂƚŝŽŶƐƌĞƚƵƌŶƚŚĞƐĂŵĞŽƵƚƉƵƚĨŽƌĞǀĞƌǇŝŶƉƵƚ

⮚&ŝƌƐƚƐƚƌĂƚĞŐǇŝƐƚŽƉĞƌĨŽƌŵƚĞƐƚƐĨŽƌĂƌĂŶĚŽŵƐĞƚŽĨŝŶƉƵƚƐĂŶĚĐŚĞĐŬŝĨƚŚĞŽƵƚƉƵƚƐĂƌĞƚŚĞƐĂŵĞ⮚DĂŝŶĚƌĂǁďĂĐŬ Ͳ ůŽǁĂƐƐƵƌĂŶĐĞŽĨƚŚĞƌĞƐƵůƚ͘

Equivalence checking (2)Equivalence checking’ (EC) for two implementationsŵĞĂŶƐƚŚĂƚƚǁŽŝŵƉůĞŵĞŶƚĂƚŝŽŶƐƌĞƚƵƌŶƚŚĞƐĂŵĞŽƵƚƉƵƚĨŽƌĞǀĞƌǇŝŶƉƵƚ

⮚^ĞĐŽŶĚƐƚƌĂƚĞŐǇŝƐƚŽƵƐĞĂĨŽƌŵĂůĂƉƉƌŽĂĐŚ͘⮚/ŶŐĞŶĞƌĂů͕ĐŽŵƉƵƚĂƚŝŽŶĂůĐŽŵƉůĞǆŝƚǇŽĨĨŽƌŵĂůĞƋƵŝǀĂůĞŶĐĞĐŚĞĐŬŝŶŐŝƐƚŚĞĐŽͲEW ĐůĂƐƐ͘⮚&ŽƌƚǁŽŝŵƉůĞŵĞŶƚĂƚŝŽŶƐƚŚĞE&;ŽŶũƵŶĐƚŝǀĞEŽƌŵĂů&ŽƌŵͿĨŽƌŵƵůĂƐ
&ϭ;yͿсzϭŽƌĂǌ &Ϯ;yͿсzϮ ƐŚĂůůďĞĐƌĞĂƚĞĚ͘⮚EĞǆƚ͕zϭ⊕ zϮ≠ 0 constraints shall be added.
⮚/Ĩ^dͲƐŽůǀĞƌƌĞƚƵƌŶƐhE^dŝƚŵĞĂŶƐƚǁŽŝŵƉůĞŵĞŶƚĂƚŝŽŶƐĂƌĞĞƋƵŝǀĂůĞŶƚ͘231

Equivalence checkingǆĂŵƉůĞ͗dĂƐŬ͗ŶĐŽĚĞ
ǆϭнǆϮнǆϯнǆϰ сϬ;ǁŚĞƌĞ нĚĞŶŽƚĞ ĂĚĚŝƚŝŽŶ ŵŽĚƵůƵ ϮͿĂƐEZĨŽƌŵƵůĂ͘^ŽůƵƚŝŽŶ͗΀;∼ ǆϭΔ∼ ǆϮΔ∼ ǆϯ Ϳсх∼ ǆϰ ΁Δ ΀ǆϭΔ∼ ǆϮΔ∼ ǆϯ Ϳсхǆϰ΁Δ͘͘͘
;ǆϭ ǀ ǆϮ ǀ ǆϯ ǀ ∼ ǆϰͿΔ;∼ ǆϭ ǀ ǆϮ ǀ ǆϯ ǀ ǆϰͿΔ͘͘͘
ϴĐůĂƵƐĞƐ ǁŝƚŚϰůŝƚĞƌĂůƐ

Equivalence checkingǆĂŵƉůĞϮ͗dĂƐŬ͗ŶĐŽĚĞ
ǆϭнǆϮнǆϯнǆϰ сϬǆϯнǆϰнǆϱ сϬĂƐEZĨŽƌŵƵůĂ͘^ŽůƵƚŝŽŶ͗ϴĐůĂƵƐĞƐǁŝƚŚϰůŝƚĞƌĂůƐĂŶĚϰĐůĂƵƐĞƐǁŝƚŚϯůŝƚĞƌĂůƐ

Equivalence checkingǆĂŵƉůĞϮ͗dĂƐŬ͗ŶĐŽĚĞ
ǆϭнǆϮнǆϯнǆϰ сϬǆϯнǆϰнǆϱ сϬĂƐEZĨŽƌŵƵůĂ͘^ŽůƵƚŝŽŶ͗ϴĐůĂƵƐĞƐǁŝƚŚϰůŝƚĞƌĂůƐĂŶĚϰĐůĂƵƐĞƐǁŝƚŚϯůŝƚĞƌĂůƐ
ĚĚŝŶŐƐĞĐŽŶĚĞƋƵĂůŝƚǇƚŽƚŚĞĨŝƌƐƚŽŶĞǁĞŽďƚĂŝŶ

ǆϭнǆϮнǆϱ сϬǆϯнǆϰнǆϱ сϬtŚĂƚ ŐŝǀĞƐ ƵƐ ϴĐůĂƵƐĞƐǁŝƚŚϯůŝƚĞƌĂůƐ

232

Formal verification of cryptoprotocolsǆĂŵƉůĞ ŽĨĐƌǇƉƚŽ ƉƌŽƚŽĐŽů
EĞĞĚŚĂŵͲ^ĐŚƌŽĞĚĞƌƉƌŽƚŽĐŽů
ůŝĐĞ

Žď

ƐŬ͕ƉŬ͕ŶϭƐŬ͕ƉŬ͕ŶϮƉŬ;ůŝĐĞ͕ŶϭͿƉŬ;Ŷϭ͕ŶϮͿƐŬ;ƉŬ;Ŷϭ͕ŶϮͿͿƐŬ;ƉŬ;ůŝĐĞ͕ŶϭͿͿƉŬ;ŶϮͿ

ƐŬ;ƉŬ;ŶϮͿͿǆĂŵƉůĞ ŽĨĐƌǇƉƚŽ ƉƌŽƚŽĐŽů
EĞĞĚŚĂŵͲ^ĐŚƌŽĞĚĞƌƉƌŽƚŽĐŽů ĨůĂǁ –ŵĂŶͲŝŶͲƚŚĞŵŝĚĚůĞ ĂƚƚĂĐŬ
KƐĐĂƌ
ůŝĐĞ

ƉŬK;ůŝĐĞ͕ŶϭͿƉŬ;ůŝĐĞ͕ŶϭͿƉŬ;Ŷϭ͕ŶϮͿƉŬ;Ŷϭ͕ŶϮͿƉŬK;ŶϮͿ

ƉŬ;ŶϮͿ

233ŽďDŽĚŝĨŝĐĂƚŝŽŶ͗EĞĞĚŚĂŵͲ^ĐŚƌŽĞĚĞƌͲ>ŽǁĞKƐĐĂƌůŝĐũĂ
ƉŬK;ůŝĐĞ͕ŶϭͿƉŬ;Žď͕Ŷϭ͕ŶϮͿƉŬK;ŶϮͿ

ŽďƉŬ;ůŝĐĞ͕ŶϭͿƉŬ;Žď͕Ŷϭ͕ŶϮͿ!

ƉŬ;ŶϮͿ

WƌŽsĞƌŝĨ ŚŽǁ ĚŽĞƐ ŝƚ ǁŽƌŬDŽĚĞůƐƚŚĞƉƌŽƚŽĐŽůĂŶĚĚĞƐŝƌĞĚƉƌŽƉĞƌƚŝĞƐĂƐ,ŽƌŶ ĨŽƌŵƵůĂ͗>ĞƚĂƚƚ;ǆͿĚĞŶŽƚĞĨŽƌŵƵůĂƐĂǇŝŶŐƚŚĂƚĂŶŝŶĨŽƌŵĂƚŝŽŶǆŝƐŬŶŽǁŶƚŽƚŚĞĂƚƚĂĐŬĞƌ͗➢ Ăƚƚ;ǆͿĨŽƌĞǀĞƌǇ ƉƵďůŝĐŝŶĨŽƌŵĂƚŝŽŶ ǆ;ŝŶĐůƵĚŝŶŐ ŝŶĨŽƌŵĂƚŝŽŶ ƐĞŶƚ ŽŶƉƵďůŝĐĐŚĂŶŶĞůͿ➢ Ăƚƚ;ĨͿΔĂƚƚ;ǆͿ схĂƚƚ;Ĩ;ǆͿͿĨŽƌĂŶǇ ĨƵŶĐƚŝŽŶ ĨĂŶĚĂŶǇ ĂƌŐƵŵĞŶƚŽĨĨ͘Ğ͘Ő͘Ăƚƚ;ƉŬͿΔĂƚƚ;ůŝĐĞͿΔĂƚƚ;ŶϭͿсхĂƚƚ;ƉŬ;ůŝĐĞ͕ŶϭͿͿ➢ /Ĩ ďǇƉƌŽƚŽĐŽů ĂŐĞŶƚƌĞĐĞŝǀĞƐ ŵĞƐƐĂŐĞ ǆĂŶĚƌĞƐƉŽŶĚƐ ǁŝƚŚƌĞƐƉ;ǆͿƚŚĞŶ͗Ğ͘Ő͘Ăƚƚ;ǆͿсхĂƚƚ;ƌĞƐƉ;ǆͿͿ/Ĩ ƚŚĞĂďŽǀĞ ĨŽƌŵƵůĂ ΔΕĂƚƚ;ƐͿŝƐ hE^dƚŚĞŶ ƐŝƐ ĂƐĞĐƌĞƚ ƚŽƚŚĞĂƚƚĂĐŬĞƌ͘

Applied Pi-Calculus
in ProVerif^ŽƉŚŝƐƚŝĐĂƚĞĚƚŽŽůƐ

ŝĨĨŝĐƵůƚƚŽƵŶĚĞƌƐƚĂŶĚĨŽƌŶŽŶͲƐƉĞĐŝĂůŝƐƚƐ

234

User-friendly automatedformal verification
platform➢ WƌŽǀĞƌŝĨ͕dĂŵĂƌŝŶĞƚ͘Ăů͘ĂƌĞĐŽŵƉůŝĐĂƚĞĚƚŽŽůƐ͕ŶĞĐĞƐƐĂƌǇĞǆƚĞŶƐŝǀĞĐŽĚĞǁƌŝƚƚĞŶďǇĞǆƉĞƌƚ

➢ KŶŐŽŝŶŐǁŽƌŬŽŶ/^Kͬ/ϮϵϭϮϴ;ŵƵůƚŝƉĂƌƚͿǁŝůůƌĞƋƵŝƌĞĂƉƉůŝĐĂƚŝŽŶŽĨĨŽƌŵĂůŵĞƚŚŽĚƐŝŶĐƌǇƉƚŽƉƌŽƚŽĐŽůǀĞƌŝĨŝĐĂƚŝŽŶ
KƵƌĂŝŵ͗ƚŽƐŝŵƉůŝĨǇƚŚĞƉƌŽĐĞƐƐŽĨƉƌŽƚŽĐŽůǁƌŝƚŝŶŐǁŝƚŚĨŽƌŵĂůƉƌŽŽĨŽĨĐŽƌƌĞĐƚŶĞƐƐ

Alice and Bob languageĂƐǇƚŽƌĞĂĚĂŶĚǁƌŝƚĞ

Building a formal bridgeůŝĐĞĂŶĚŽď>ĂŶŐƵĂŐĞ

ƉƉůŝĞĚWŝͲĐĂůĐƵůƵƐ
/ŶWƌŽsĞƌŝĨ

235

Practical goals

Translator from AnB languagewith cryptographic primitives to Proverif’s Pi-calculus

Use Case: EAP-AKAEƵŵďĞƌŽĨůŝŶĞƐ ŽĨĐŽĚĞĐŽŵƉĂƌŝƐŽŶ
Protocol\Language

Alice and Bob

Proverif

Simple Example

15

40

Needham-Schoeder

18

54

Otway-Rees

21

64

236

Practical goals

Translator from AnB languagewith cryptographic primitives to Proverif’s Pi-calculus
Formally verified translator

F. Montesi “Choreographic Programming”
F. Montesi “Introduction to Choreographies”

Properties verified
Formally verified* AnB to ProVerif translator with automatic checks:➢ ^ĞĐƌĞĐǇ
➢ &ŽƌǁĂƌĚ ƐĞĐƌĞĐǇ
➢ tĞĂŬƐĞĐƌĞĐǇ
➢ /ŶĚŝƐƚŝŶŐƵŝƐŚĂďŝůŝƚǇ➢ ƵƚŚĞŶƚŝĐĂƚŝŽŶ➢ EŽͲƌĞƉůĂǇƵƚŚĞŶƚŝĐĂƚŝŽŶ
➢ <ĞǇ ŽŵƉƌŽŵŝƐĞ /ŵƉĞƌƐŽŶĂƚŝŽŶ

ƵƚŽŵĂƚŝĐĨŽƌŵĂůĐŚĞĐŬƐŚĂŶĚůĞĚďǇĚŽĐŬĞƌŝǌĞĚ WƌŽǀĞƌŝĨ͕ĂǀĂŝůĂďůĞŽŶŽƵƌWůĂƚĨŽƌŵ
ΎtŽƌŬŝŶƉƌŽŐƌĞƐƐ

237

Challenges*:➢ Formally verifiable translations to other protocol languages➢ Extended list of security properties* currently under consideration

Thank you238

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Synergies of medicine, physics, and mathematics inmedical imaging
Shizuo KajiInstitute of Mathematics for Industry, Kyushu University, Japan
Medical imaging provides detailed visual representations of internal structures andfunctions of the human body and plays a pivotal role in diagnosing, monitoring, andtreating various medical conditions. Mathematical disciplines intersect with medical
imaging in multifaceted ways, encompassing:• Image reconstruction involves the transformation of raw measurements acrossdiverse modalities such as computed tomography (CT), magnetic resonanceimaging (MRI), and ultrasound into coherent, human-interpretable images.
• Image enhancement and information Extraction aim at refining image
quality while extracting vital information embedded within.
• Quantitative analysis unveils deeper insight into the heterogeneity and progression of diseases in an objective and reproducible manner.
We will present some of our collaborative endeavours, bridging the expertise of medical
doctors, medical physicists, and the realm of mathematics. Our work showcases applications of machine learning and topology that fortify and enrich the field of medical
imaging.

239

SYNERGIES OF MEDICINE, PHYSICS,
AND MATHEMATICS IN MEDICALIMAGING
WORKSHOP on Mathematics for Industry
BASIS OF MATHEMATICS IN NANOMEDICINE STRUCTURES AND LIFE SENSING
25-29 SEPTEMBER 2023 (WARSAW)
Shizuo KAJI (IMI, Kyushu U.)

Applied-Applied
Mathematics

Physics

Application!

Applied
Maths

Pure
Maths

Medicine

Application

Moebius Kaleidocycle, 2018

S. Kaji, J. Schoenke, E. Fried, M. Grunwald

Kyoto U. hospital & U. Tokyo hospital & Kyushu U. IMIA collaborative project on various aspects of medical imaging
Nuclear physics
Inverse problem
Scientific ML
Measurement

Clinical Data
Topological Invariants
Machine learning

Biomarkers
Simulation
FeatureExtraction

Reconstruction

Diagnosis
Prognosis

Image from Wikipedia
Figures from J. Appl Physiol, vol 131-2, 2021

Evaluation
Developing clinically useful methods often leads to theoretically interesting questions

They look locally similar,
but we see a clear difference if we zoom out

c.f. Manifolds are locally all Euclidean and homology distinguishes the global topology of them.

240

DL is good at• Precise observation• Memorising/imitating examples• Processing huge data
• Accurate operation

Human is good at• Rough estimation• Panoramic view
• Discovering rules/invariancefrom a small number of examples• Explaining the reason

Deep Learning(DL)
Data-driven
local

Topological Data Analysis(TDA)
Maths-based
global

enta

plem

com

ry

Background• DL achieves high performance but has some weakness• TDA has been proven effective in capturing data
features that conventional techniques have missed

Image Feature Extraction

localRadiomics

Usually 2D

Convolutional
Neural Network

High computational cost for 3D

Deep-learning based

Attention-based
Neural Network

Requires a large dataset for training
High computational cost for 3D

TopologicalInvariants

Is not versatile but complements
neural-network-based methods
globalRemark: Deep Learning is REALLY powerfulwith hematologists at Tokiwa hospitalwith physicists at U. Tokyo hospital

Detection ofabnormal blood cells

CBCT

Synthesized PlanCT

PlanCT

Low exposure
Low image quality

dilemmaHigh exposure
High image quality(Kida, Kaji, et al., Med. Phys., 2020)(Mori, Kaji, et al., Sci. Rep., 2020)

Diagnosis

Reconstruction241Remark: DL and TDA are NOT competitors but collaborators
Example: Sinkhole detection in ground penetrating radar image(with S. Choi, T. Kim, GK Engineering)

Difficulty
1. No big data (data acquisition is costly)
2. False Negative (overlook) is criticalFP

correct

FN

By combining CNN and TDA, we achieved comparable performancewith human experts with only 40 labelled volumes
Why it worked well: Topology helped to distinguish sinkholes from pipes, which were reflected in H1

Topological Featuresof Image

Pipeline

Filtered complex

2D/3D image

Sequence of
topological objects

Topological invariants
Features

Results+Interpretation

Domain knowledge +Machine learning242

Image = Function on a metric space

A colour image is represented by atriple of real-valued functions (R,G,B)
We focus on a monochrome image
𝑓𝑓: 𝑋𝑋 → 𝑅𝑅 (𝑋𝑋 ⊂ 𝑅𝑅! , 𝑛𝑛 = 2,3)

An image processing/analysis method is
an operation on the space of functions

Image processing = Operation on functions
image

image

Example:Convolution

Example of a kernel:

Topological Image Analysis

Function è Space è “Numbers”
topological space X
function 𝑓𝑓: 𝑋𝑋 → 𝑅𝑅
Each threshold value
gives rise to
the sub-level set{x | f(x)<a}243

Function è Space è “Numbers”
For each threshold a, we have a spaceX(f,a) := {x | f(x)<a}
We can compute topological invariantsof X(f,a) to obtain image features

Q: How to choose a threshold?
A: We do not choose. Use them all!

Under review as a conference paper at ICLR 2023

Persistent homology (PH)

◦Extension of homology deﬁned forfunctions over topological spaces
parameters of the image generation. Several different formuli are tested and an iterated f
Under review
a conference paper
at ICLR 2023 the◦Foreach astopologicalfeature(cycle),
system,which generates fractal images, is found to be effective. A wider variety of image gen
threshold values withwhich
it wasborn
methods havebeen
tested since then (Baradad et al., 2021; Kataoka et al., 2022). To see how s
and destroyed are recorded
images are helpful in acquiring image features is interesting also in terms of cognitive scienparameters of the image generation.
Several different
formulitested andimages,
an iterated whichfunction are not very meaningful for human ey
In this paper,
we alsouse aresynthetic
system, which generates fractal images, is found to be effective. A wider variety of image generation
Remark:
try
to
learn
topologicalfeaturesfromthem.
One technical difference from the Formula
methods
beensincepaper
then(Baradadet al., 2021; Kataoka et al., 2022). To see how synthetic
Under
a conferenceat
ICLRof2023
We canreview
alsohaveviewasPHastesteda “continuous
relaxation”
homology.
imagesare helpful
in quantity
acquiring
image
featuresis interestingalso interms
of cognitive
science.
Supervised
Learning
is howthelabelsare generated.
Instead of fixed labels associated with th
Homology
is a discretethat issometimes
problematic.
(e.g., homology can change abruptly with small variation in the input)

In this paper, we also use generation
synthetic images,
whichparameters,
are not very meaningful
for humaneyes, byanda mathematical formula computed directmodel
we generatelabelstry to learn topological featuresfrom them.
One technicaldifference fromthe Formula-driven
the images,
which
has advantage
of being
applicable to any image generation model. Th
Supervised Learning is how the labels are generated. Instead of fixed labels associated with the image
encodetopologicalfeaturesin our paper) and the model is encouraged
parameters
the imagegeneration.
Severaldifferent
formuli are(persistenttestedandanhomology,
iteratedfunction
generation ofmodelparameters,
we generatelabelsby a mathematicalformulacomputed
directlyfromsystem,whichwhich
generates
fractalimages,is foundtobe effective.
A widervariety
ofmodel.
image generation
the images,
has advantage
offeatures
being
applicable
to anyimage
generation
The thelabelstopological features.
image
that
are
relevanttoapproximate
methods
havebeentestedsince
then(Baradad
et
al.,
2021;
Kataokaetal.,2022).
Tosee
howsynthetic
encode topological features (persistent homology, in our paper) and the model is encouraged to learn
imagesare helpful
acquiring
features istheinteresting
alsofeatures.
in terms of cognitive science.
image features
thatinarerelevantimage
to approximate
topologicalIn this paper, we also use synthetic
which areHOMOLOGY
not very meaningful
forIMAGE
human eyes,
and
2.3 Pimages,
ERSISTENTFORANALYSIS
2.3to learn
P ERSISTENTHOMOLOGYFOR IMAGE
try
topologicalfeatures fromthem. ANALYSIS
One technical difference from the Formula-driven
Supervised Learning is how the labels are generated. Instead of fixed labels associated with the image
hasit isanseen
unusualinput
andoutputPersistent model
homology
has anPersistentunusualinputhomology
and output
when
as a feature
extractor.
PHtakes when it is seen as a feature extractor. P
generation
parameters,
we generatelabelsby a mathematicalformulacomputed
directly
froma seriesof nested
spacesa multisetof imageintervalsof the real
numbers.
theimages,whichtopologicalhas advantage
of andbeing
applicable
to anygeneration
model.
TheWe
labelsa series
ofoutputs
nestedtopologicalspacesandoutputs
aview
multiset of intervals of the real numbers. W
an imageas a functionf : (persistent
X ! R definedover ainrectangulardomainX, andis we
obtain a seriesofencodetopologicalfeatureshomology,
our paper)
and
the!
model
encouraged
to learn
an
image
as
a
functionf:XR
definedovera rectangular domain X, and we obtain a s
nestedfeatures
spaces that are relevant to approximate the topological features.
image

Persistent nested
homology
spaces (formal definition)

⇢

⇢

⇢

⇢

=

⇢

=

t1
t2 sequence
tm of F -vectorti
PH is by definition
thespaces (for each dimensioni d)2⇢; ⇢ Xt1 ⇢ Xt2 ⇢ · · · ⇢ Xtm = X, Xti = {(x, y) 2 X | f (x, y)  ti },2.3 P ERSISTENT HOMOLOGY FOR IMAGE ANALYSIS
where
tm = sequencemax(f ). of
Applying
the X
coefficientsin theXti = {(x, y) 2 X | f (x, y)  ti },Increasing
spacesthe homology; ⇢ Xtfunctor⇢ XHtd2(−)⇢ with··· ⇢tm = X,
1
field F2 homology
with twohas
elements,
weinput
obtaincorrespondingof extractor.
F2 vector
Persistentan unusual
and theoutputwhen it is seensequence
as a feature
PHspaces,
takes
this
sequence
is by wheredefinition
thehomology
P Hof
) the
of
thehomology
pairWe(X,f ). functor H (−) with the coefficients
Apply
thepersistenthomologyfunctord (X,aandseriesof nested
topologicalspacesandoutputs
amax(fmultisetofintervals
the freal
numbers.
view
t
=
).
ApplyingmdPHf )acanbe written
direct coefficientssum
so-calledinterval
modulehavinga series
the formd (X,asanimage
functionf : X as! the
R defined
over aofrectangulardomainX, andwe obtainof(within F2) the
field
F
with
two
elements,
weobtainthecorresponding
sequence of F2 vector2
nested0 spaces· · ·
0
F2
···
F2
0
···
0
and this sequence is by definition the persistent homology P Hd (X, f ) of the pair; ⇢ X ⇢ X ⇢ · · · ⇢ X = X, X = {(x, y) 2 X | f (x, y)  t },

⇢

⇢

⇢

⇢

=

P Hd (X, f ) can be written as the direct sum of so-called the interval module having thwhere
Htjd (−)
withthetjcoefficientsin
Hd (;)tm =· · ·max(fHd).(XtApplying
Hthe· · · functor
Hd (X
Hd (X
)
···
Hdthe
(X).
d (Xhomology
ti )
i 1)
1)
0 obtain[t·ithe· · When d =00, itsequence
···
F2
0
···
field
F
with
two
elements,of FF
spaces,2
2 2vector
We denote the summand by theweinterval, tj ).corresponding
may happen
that
j−1 =
m, in
and
thiscasesequence
is by thedefinition
thebypersistenthomology
PHf ) ofofthe
pair
(X,
f ).
which
we represent
summand[tdirectFig.
1“intervals”
forand (X,
examplePHtheof animage.
The
sequence
decomposes
into
the
sum
ofthe
havingform
i , 1). See
PTo
Hdsum
(X, up,f ) canbewrittenas
thedirectsumofso-calledinterval
modulehavingtheform
PH takes a function f : X !IdR and outputs
a multiset of intervals. The alien output
Id· · be transformed
0· · · vector F
0 techniques···02
as 0multiset·can
into Fa 2fixed-length
using
vectorisation(Adams
H
(;)
·
·
·
H
(X
Hd (X
···
Hd (Xtj 1 )Hd (Xtj )···etwhichal., 2017;
Bubenik, 2015;
Chungin)the standard
machined & Lawson, 2021) so thatd it tfitst ) learningi 1
correspondto cycles
represented by 𝑡𝑡i" , 𝑡𝑡# ∈ ℝ$
pipeline.
(=
generators=
topologicalfeatures)
When d = 0, it may happen that j − 1 =Hd (;)···Hd (Xti 1We) denote
Hd (Xti ) the· ·summand·
H (Xtj by) the
Hdinterval(Xtj )· ·[t· , tHj ).
d (X).
CNNs
tend to be biased
towards texture
when traineddwith
a1 classification
task of inaturalimwhichcasehuman
representsummandby[tWe
the etsummandbythecontrast,
interval[tiwe, tj ). perception
When d =relies
0,theit mayhappenj −shape
1 i=, 1).
m,
in See Fig. 1 for an example of PH of anagesdenote(Geirhosal., 2019).
In
muchon thethat
globalof thewhich
we represent
the
by
[tPH
Seea Fig.
1 for
an exampleof PH
image.
imagecasecontent.
Therefore,
itsummand
would
beneficial
for
model
to learn
globaltopologicalfeaturesi , 1).
To
sumbeup,takes
a function
f :X!ofRanandoutputs a multiset of intervals. The alienTo
PH takesa function: X ! R by
andanoutputs
a multiset
of intervals.
ThePH
alien
outputas sumwellup,as localfeatures.
This isf supported
experimentin whichcombining
with
other
as
multisetcanbetransformed
intoafixed-length
asdescriptors
multiset canbe transformed
intoperformancea fixed-length
vectorrecognition
using vectorisation(Adamsresults
in an increasedin object(Li et al.,techniques2014), showingthat vector using vectorisation techniques (etPHal.,provides
2017; Bubenik,
2015;etChung&
Lawson,2021)
so that
it fits ones.
intheThere
standard
machine studies
learningal.,2017;
Bubenik,
2015;
Chung
&areLawson,2021)
a shape featurethatis complementary
to conventional
previous
to so that it fits in the standard machine lpipeline.
assimilate PH into deep learning.
To deal with PH with neural networks, a parametric representation
pipeline.
of
persistencehomology
withlearnableparametersisintroduced
in
Hofer
et
al.
(2017)
sothat
CNNs tend to be biased towards texture when244
trained with a classification task of natural im-a
task-optimal
in humana data-driven
manner.
Dedicatedof CNNsareages(Geirhosvectorisation
et al., 2019).is
Inobtained
contrast,
perception
reliesmuch onarchitectures
theglobal shapeof the
CNNs
tend
to
be
biasedtowards
texturewhen
trained with a classification task of natu
designedin Somet al. (2020)
for computing
vectorised
PH oftotime-series
andpoint clouds
in theimage
content.
Therefore,
it wouldbe beneficial
for a model
learn global
topologicalfeaturesages(Geirhosetal.,2019).
In
contrast,
human
perception
relies much on the global shap
form
of
the
persistence
image.
The
topologicalautoencoder
(Moor
et
al.,
2020)
learnsalatentspace
as well as local features. This is supported by an experiment in which combining PH with otherof a point cloudpreservestheperformance
topologicalterms ofitpersistent
Our idea
is a model to learn global topological f
image
content.structurewouldbebeneficial
descriptors
resultsthat
in anincreasedin Therefore,
objectinrecognition(Li et al.,homology.
2014),
showing
thatfor⇢⇢⇢⇢2D Example

Figure 1: A greyscale image, considered as a function f on a rectangular grid, defines a growing
A cyclethe form
PHby
0=sequence of spacesits{(0,1],
sublevel(0,∞)}sets. The(islands)
shaded regions indicate
theofsublevelset(a,b]
Xt is=represented{(x,$ y) |
𝑎𝑎, 𝑏𝑏 of∈ 𝑅𝑅
a point
a 1thresholdt. The degree
0 persistent homology isbythe(multi)setintervalsf (x, y) < t} forPH= {(1,2]}(holes){[0, 1], [0, 2]}. For example, the interval [0, 1] corresponds to the connected component (“born” at
t = 0) consisting of the single pixel at the bottom-left corner in the left-most image, which disappears
in the central image (“killed” at t = 1). The degree 1 persistent homology is {[1, 2]} whose element
represents the hole surrounding the two pixels with the value 2 in the centre image. This holedisappears in the right-most image. Since the space is two-dimensional, persistent homology is
non-trivial only at degrees 0 and 1. In this way, given a growing sequence of spaces indexed by real
numbers, persistent homology records the topological features, islands and holes, with the indices inwhich the features emerge and disappear.

90
91
92
93
94
95
96
97
98

99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121

and its variants learn a compression-expansion task; in other words, it aims at obtaining a complete
low-dimensional feature of an image that can recover the image. Since knowledge in general meansa simple and compressed description of a target object, compression-expansion can be considered
as a fundamental task in learning, and the learned features are useful for many downstream tasks.
Another popular framework of SSL is the contrastive learning (see [15] for a survey). In this scheme,◦ Cubical
RipserK-Sudo-Ahara,
a low-dimensional
embeddingof (images
is learned2021)
so that the original image and its perturbation are◦ Open-source(MITlicense),Available
my githubrepository
placed nearby whereas
different
imagesare placed
far atapartin terms
of a certain metric defined on thehttps://github.com/shizuo-kaji/CubicalRipser_3dim/
representation space.
Contrastive learning tasks are mostly based on human perception; for example,
what images should
be regarded“similar”
is definedby restricting
the classof image
perturbation.
◦ Capableof computing
persistenthomologyof time-series,
image,
volumetricdata◦ One of the fastest program for computing persistent homology of cubical complexes
2.3 Learning◦ withsynthetic
imagesThe only
programwhich can handle two major constructions of cubical complexes◦ Pythonthat works
nicelywith Numpyof(including
DICOM
Even though SSL
saves binding
the annotation
costs,
the preparation
training datais stillconverters)
a vexing problem.
◦ Tutorial (run on
GoogleColab): google“shizuowhich
TDA aretutorial”
For practical conveniences,
there
are ready-to-use
modelspretrained with large-scale
datasets. However, there is a concern on how the pretrained model is obtained, which is often out ofsets by sweeping thresholds
Persistent
our control. The dataset and its labels used forSublevel
the pretraining
may have been low-quality,
subject
to bias, and violated
and privacy. For example, ImageNet, one of the most popular
2D orusage
3D rights
image
homology
large-scale datasets, suffers from fairness issues [16], and there have been a growing interest in thefairness of machine learning [17]. For providers of pretrained models, adversarial actions such as themodel inversion attack [18] are also problematic, which would reveal sensitive information in the data
used for pretraining. No matter how much care is paid for data collection, it is impossible to be freefrom all kinds of these issues as long as real images are used. Using generative adversarial networks(GANs) to generate image datasets for training is a popular and successful strategy ([19]) to mitigate
the situation, but GANs are also trained with natural images and cannot avoid above-mentioned
problems. A promising approach is to use algorithmically synthesised images. Formula-driven
Supervised Learning introduced in [20] considers pretraining with synthetic images generated by amathematical formula. The labels are assigned according to the parameters of the image generation.
Several different formuli are tested and an iterated function system, which generates fractal images, isfound to be effective. A wider variety of image generation methods are tested in [21]. Using synthetic
images to acquire image features is interesting also in terms of cognitive science.

Software for Persistent Homology computation
for image and volumetric data

PH as a feature

In this paper, we also use synthetic images, which are not very meaningful for human eyes, and try
Diagramto learn topologicalfeaturesfrom them.
The main differenceOutput：Persistencefrom the Formula-driven Supervised
Input:
Function(over a topological space)
Learning is how the labels are generated. Instead of fixed labels associatedwithpoints
the image(finitein generation
R2)
model parameters, we generate labels by a mathematical formula computed directly from the
1D

3

Red: PH0 cycles
Blue: PH1 cycles

2D, 3D

245

Applications
to CT analysis
We cannot include the details here
since some materials are not publishable online.
Contact Kaji for a full copy of the slides.

Summary
Topology (persistent homology) provides a way to extract image/volume
features that are not easy to obtain by conventional method
Global and invariant features encoded by persistent homology (PH)complement those (mainly local) features obtained by deep learning (DL)and can be used in conjunction to boost performancePH-based image analysis has some advantages:

robust and easily transferable (ó DL needs re-training)interpretable (ó DL is often a blackbox)
3D (ó many conventional analyses are 2D slice-based)

246

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Plasticity – Modeling and mathematical analysis
Konrad Kisiel
Faculty of Mathematics and Information Science,Warsaw University of Technology, Poland(joint work with Krzysztof Chelmiński)
Systems of equations describing an inelastic response of metals, with the fundamental
assumption of small deformations, consist of linear partial differential equations coupledwith nonlinear differential inclusions (or ordinary differential equations) for the vectorof internal variables. The partial differential equations result from general mechanicallaws. The differential inclusions are experimental, and depend on the kind of consideredmaterials. One of the main assumptions needed in known existence theories is so-calledsafe-load condition. This kind of assumption is an indirect assumption on regularity ofdata. Our main goal is to present a method to obtaining existence of solutions, where
the safe-load condition can be replaced by an assumption abouth the size of the set ofaddmissible stresses.

247

Plasticity – Modeling and mathematical Analysis
Krzysztof Chelmiński, Konrad Kisiel
Faculty of Mathematics and Information Science
Warsaw University of Technology

Workshop on Mathematics for Industry
Warszawa, 25-29 September, 2023

1

Table of contents

1. Theory of inelastic deformations - short introduction
2. Elasto-perfect plasticity
3. Safe-load condition
4. Energy estimates without safe-load condition

2

1. Theory of inelastic deformations - short introduction
Let Ω ⊂ R3 be a bounded domain with a smooth boundary ∂Ω .
Balance of momentumρutt(x, t) = divx T (x, t) + F (x, t) ,ρutt ∼ 0 (quasistatic case)(u, T ) : Ω × (0, Te) → R3 × S 3 – (the displacement vector, the stress tensor)
F : Ω × (0, Te) → R3 – the given external force, ρ > 0 – the mass density
Elastic constitutive relation
T (x, t) = D ε(x, t) − εp(x, t)ε = 12 (∇u + ∇T u) – the linearized strain tensor
εp : Ω × (0, Te) → S 3 – the plastic strain tensor, D : S 3 → S 3 – the elasticity
tensor (symmetric, > 0)
Inelastic constitutive relation
εpt(x, t) ∈ f ε(x, t), εp(x, t)

f : D(f ) ⊂ S 3 × S 3 → P(S 3) – a given constitutive multifunction
3248

Models of premonotone type
Prof. Dr. Dr. h.c. Hans-Dieter Alber in the monograph Materials with memory
LNM 1998 has defined a very large class of models: models of premonotone type.
Definition 1
A model is called of premonotone type if the inelastic constitutive relation is in theformεpt ∈ g − ρ∇εp ψ(ε, εp)where ψ(ε, εp) = 12 D(ε − εp) · (ε − εp) is the free energy function andg : D(g) ⊂ S 3 → P(S 3) is a given inelastic multifunction satisfying:∀z ∈ D(g)g(z) · z ≥ 0(∗)

If we additionally assume that g(0) ∋ 0 (∗) ⇔ monotonicity at the point 0. All
models used in practice are of premonotone type.
Models of monotone type ⇔ g is additionally monotone
4

2. Elasto - perfect plasticity (the Prandtl–Reuss model)
εpt(x, t) ∈ ∂IK T (x, t) ,K = dev K × {c · I : c ∈ R}

where dev T = T − 1/3 (tr T ) · I. Moreover, dev K is convex with 0 ∈ int (K) .
Hencky flow rule

..
...
....
.
...
....
.
.
....
...
..
...
....
....
..
...
.....
...
....
.....
.
..
....
...
..
....
...
...
...
...
..
.....
....
..
...
...
..
....
...
...
...
...
..
....
....
...
...
...
..
.....
....
..
...
...
..
....
...
...
...
...
..
....
....
..
...
...
.................................................................................................................................................................................

dev K = B(0, k)⇔∀ S ∈ K |dev S| ≤ k .

..

.
....
.
|εpt | ........



k



|dev T |


c·I

...
....
.
...
....
...
..
.....
..................................................................................................
....
.............
...................
..
..........
.............
...
...
........
..........
..
.......
...
........
...... ...
.... ............
..... ...
... .......
... ..
.....
........
..
...
...
.....
.......
.
.
............
.
... .
...... ....
... ............
.......
........
..
...
........
.
..........
.
.
.
.
.
.
.
....
.....
.
..
.............
.............
...................
..
...
...................................................................................................
...
...
....
....
.
.
...
...
...
...

dev K

K

S ∈ ∂IK (T ) ⇔ (S, T − τ ) ≥ 0 ∀ τ ∈ K∂IK (T ) is monotone and 0 ∈ ∂IK (0)5

3. Safe-load condition
Definition 2 (quasistatic case)∗such that the
The given data F, gN satisfy the safe-load condition if there exists gD
unique solution (u∗, T ∗) of the linear system

divx T ∗(x, t) = −F (x, t)
T ∗(x, t) = Dε(u∗(x, t))∗u∗(x)|ΓD = gD(x, t) , T ∗(x) · n(x)|ΓN = gN (x, t) .
have the regularity:u∗ ∈ W1,∞(H1), T ∗ ∈ W1,∞(L2) and there exists δ > 0 such that{T ∗ + σ : |σ| ≤ δ} ⊂ D(g)and there exist uniformly bounded in L∞(L2) selections of the sets g(T ∗ + σ).
For the Prandtl-Reuss model with the Hencky flow rule this condition is equivalent
to: there exists δ > 0 such that |dev T ∗| ≤ k − δ .
6249

Theorem 1
k
If the given data satisfy the safe-load condition then the sequences {εp,kt } , {εt } froma “good enough” approximation are bounded in the space L∞(L1).
Remark 1
Without any additional geometrical conditions for g the strains are weakly relatively
compact in the space L∞(M) where M is the space containing bounded measures.
Remark 2
C. Johnson in 1976 was the first mathematician, which has formulated the safe-loadcondition for the Prandtl-Reuss model. The condition of Johnson is a little bit weakeras presented in this lecture.
The Johnson safe-load condition for the Prandtl-Reuss model
There exists a stress field S ∗ such that−div S ∗ = F , S ∗·n = gN and ∃ δ > 0

S ∗ + B(0, δ) ⊂ K ⇔ |dev S ∗| ≤ k − δ .

7

4. Energy estimates without safe-load condition
Let us consider for simplicity the quasistatic Prandtl-Reuss model .
−divxT = F,
T = D (ε − εp) ,εpt ∈ ∂IK (T ) ,Our approach is to modify only the inelastic constitutive equation and consider thefollowing problem−divx T λ = F,T λ = D ελ − εp,λ ,
 λ
p,λ
λ
εt = M T ,where Mλ : S 3 → S 3 denotes the Yosida approximation of the maximal-monotone
operator ∂IK .
8

Let us recall the definition of the space LD(Ω).
Definition 3






LD (Ω) = u ∈ L1 Ω; R3 : ε (u) ∈ L1 Ω; S 3

LD (Ω) is the Banach space equipped with the standard norm∥u∥LD(Ω) = ∥u∥L1(Ω) + ∥ε (u)∥L1(Ω) .
Theorem 2

Assume that Ω ⊂ R3 is open, bounded and ∂Ω ∈ C 1. Then, there exists a boundedlinear operator
γ : LD (Ω) → L1 ∂Ω; R3 ,such that γ (u) = u|∂Ω for every φ ∈ LD (Ω) ∩ C 0 Ω . Hence∥γ (u)∥L1(∂Ω) ⩽ CLD ∥u∥LD(Ω).

∃ CLD > 0 ∀ u ∈ LD (Ω)

Moreover, the following embedding theorem holds,∥u∥L3/2(Ω) ⩽ CELD ∥u∥LD(Ω).

∃ CELD > 0 ∀ u ∈ LD (Ω)
9250

We observed that in order to obtain proper energy estimates it is enough to assume
the admissibility of the Neumann boundary data and the external force, which means
Definition 4 (Admissibility of forces)
We say that in the dynamical case the Neumann boundary data gN is admissible if
CLD ∥gN ∥L∞(0,Te;L∞(ΓN )) < C ∗,where CLD is a positive constant from the trace theorem in the space LD (Ω). The
constant C ∗ depends on the maximal monotone inelastic multifunction only (for the
Prandtl-Reuss model with the Hencky flow rule C ∗ is equal to the yield constant k.)
We say that in the quasi-static case the Neumann boundary data gN and the external
force F are admissible if
CELD ∥F ∥L∞(0,Te;L3(Ω)) + CLD ∥gN ∥L∞(0,Te;L∞(ΓN )) < C ∗,where the constant CELD is from the embedding theorem for the space LD(Ω) andthe constant C ∗ is the same as in the dynamical case.
10

Theorem 3 Assume that the data are regular enough and boundary data gN is
admissible or in the quasistatic case gN and F are admissible. Then there exists a
positive constant C, independent of λ, such that in the dynamical caseE uλt, ελ, εp,λ ,

t 
0

Ω


  
 
λ
λ λ p,λεp,λ, εp,λt · T , E utt , εt , εtt  ∞L (L1 )

≤ C.

 

where 2E uλt, ελ, εp,λ = Ω(ρ|uλt|2 + D(ελ − εp,λ) · (ελ − εp,λ)) dx
and in the quasistatic caseE ελ, εp,λ ,

t 
0

Ω

  

 
λ
λ p,λ
, εp,λεp,λt · T , E εt , εtt  ∞L (L1 )

≤ C.

 

where 2E ελ, εp,λ = Ω D(ελ − εp,λ) · (ελ − εp,λ) dx
10Literature used in the lecture• H.-D. Alber – Materials with memory – Lecture Notes in Math. v. 1682, Springer,
Berlin Heidelberg New York, 1998
• C.Johnson – Existence theorems for plasticity problems – J. M. P. Appl. 1976

• R. Temam – A generalized Norton-Hoff model and the Prandtl-Reuss law ofplasticity – ARMA, 1986
• K. Ch. – Coercive approximation of viscoplasticity and plasticity – Asymptotic
Anal., 2001
• K. K., – Dynamical problems in the theory of inelastic deformations – PhD thesis,
2018• K. K., K. Ch. – On strong solutions of viscoplasticity without safe-load conditions– J. Diff. Equ., 2020
• K. K, K. Ch. – Prandtl-Reuss dynamical elasto-perfect plasticity without safe-loadconditions – Nonlinear Analysis TMA, 2020
• K. K, K. Ch. – Quasistatic viscoplasticity without safe load conditions – J. Diff.
Equ., 2021
11

251

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Developable surfaces with curved folds andapplications
Miyuki Koiso
Institute of Mathematics for Industry, Kyushu University, JapanA developable surface is a surface which is isometric to a planar region, that is, there exists a continuous bijective mapping from the surface to a planar region which preserves
the length of every curve. If the considered surface is smooth, then it is developable
if and only if its Gaussian curvature vanishes everywhere. Moreover, in this case, thesurface can be continuously and isometrically deformed until the planar region. In thistalk, we discuss developable surfaces with curved folds, which are naturally appear
as origami works and have many applications in manufacturing objects. We discuss
intrinsic and extrinsic singular points (such as vertices and points in edges), curvaturesat each singular point, and the existence and nonexistence of continuous isometric deformations from such a surface to a planar region. We also discuss applications anddiscretization of these objects.

253

WORKSHOP on Mathematics for Industry 2023

Developable surfaces with curved folds
and applications*
Miyuki Koiso （Kyushu University, Japan）
Collaborated with:
J. Mitani (information science), T. Homma (architecture),
Y. Yokosuka (architecture), T. Kitahata (physics),
M. Yasumoto (discrete geometry), Y. Jikumaru (geometry)
September 26, 2023, Warsaw University of Technology, Poland
M1

E1

P

*This work is supported by JST CREST Grant Number JPMJCR1911
M2
E2
and JSPS KAKENHI Grant Number JP20H01801.

M3

Plan of the talk

E

E5

E3

M1

M5

E4
1

M4

M5

M2 P
We consider oriented piecewise smooth (PW smooth)M4"
M3
surfaces 𝑀𝑀 = ∑! 𝑀𝑀! in 𝔼𝔼 . Here 𝑀𝑀 is a 2-dimensional manifold,each 𝑀𝑀! is a smooth surface with boundary, and locally the
number of 𝑀𝑀! is finite.
𝑀𝑀 M1 E5 M5
 Developable surfaces
P
E1
E4
 A specific class of PW smooth developable
M2
E2 E3
M4
surfaces called “pillow boxes”, and a
M3variational problem for them.

A double flat rectangle

P1
P5

Bend, andfold along curves
A pillow box

 Continuous isometric deformations from pillow boxes to planar
regions
 Application2

Developable surfaces
Def. 1. A PW-smooth surface 𝑀𝑀 is said to be
developable if it is isometric to a planar region 𝑅𝑅 (thatis, there exists a continuous bijective mapping 𝐹𝐹 from𝑀𝑀 onto 𝑅𝑅 that preserves the length of each curve).
𝑀𝑀

F

𝑅𝑅

Remark 1. It is well-known that a smooth surface 𝑀𝑀 isdevelopable if and only if the Gaussian curvature
𝐾𝐾 𝑝𝑝 of 𝑀𝑀 vanishes at any point 𝑝𝑝 ∈ 𝑀𝑀.
254

P4

3

P8

P3
P2
P6

P7

Smooth developable surfaces（smooth surfaces with 0-Gaussian curvature）
Fact 1. Smooth developable surfaces in 𝔼𝔼" are the following:(1) cylinders, (2) cones, (3) tangent developable surfaces.
𝑃𝑃

𝛤𝛤#

𝛤𝛤#

cylinder

cone

𝛤𝛤#

tangent developable surface

Since developable surfaces can be constructed by bending
a flat sheet, they are important in manufacturing objects
from sheet metal, cardboard, and plywood (consists ofthree or more layers of veneer).

4

A variational problem for developable surfaces“Find the optimal pillow box ! ”

What is a pillow box?

A double flat rectangle
Fold(topologically, 2-spere 𝑆𝑆 $)along curves
made of paper

Pillow box
Def. 2 (Pillow box)．A pillow box is a compact PW-smooth surface
without boundary with genus 0 that consists of four parts of(generalized) cylinders and that is isometric to a double rectangle.

Q: For a given double rectangle, find the pillow box withthe maximal volume. A: We will give a (rigorous) answer.
𝑅𝑅

𝑄𝑄"
𝑎𝑎

𝑄𝑄!

𝑄𝑄$

isometric

𝑎𝑎 𝑄𝑄#

A double rectangle

𝑃𝑃"

𝑃𝑃!

𝑀𝑀"
𝑀𝑀!

𝑃𝑃#

𝑀𝑀$

𝑀𝑀#

𝑃𝑃$

Pillow box

5

Existence and uniqueness of the optimal pillow box
Theorem 1 (K)：For any given double rectangle 𝑅𝑅(2𝑎𝑎, 2𝑏𝑏) withside lengths 2𝑎𝑎, 2𝑏𝑏 (see the picture below) there exists a uniquepillow box 𝑀𝑀(2𝑎𝑎, 2𝑏𝑏) (which we call the optimal pillow box) that
encloses the largest volume. It has an explicit representation
using elliptic integrals. It consists of four (generalized) cylinders(of 𝐶𝐶 )class) of which the base curves (the top and the bottom
half of Γ# and two blue curves in the picture below right) arecongruent and they are elastic curves. Remark 2.

𝑅𝑅(2𝑎𝑎, 2𝑏𝑏)

−𝑎𝑎

𝑎𝑎

A doublerectangle

2𝑏𝑏

isometric

𝑀𝑀(2𝑎𝑎, 2𝑏𝑏)

Γ%

Pillow box

255(1) lim 𝑀𝑀 2𝑎𝑎, 2𝑏𝑏 = a
1→)right circular cylinder withradius 2𝑎𝑎/𝜋𝜋.
(2) lim 𝑀𝑀 2𝑎𝑎, 2𝑏𝑏 = two3→)parallel rectangles withwidth 𝑏𝑏 and infinite length.
6Outline of the proof of Theorem 1 (I) --- Step 1--𝑅𝑅(2𝑎𝑎, 2𝑏𝑏)

A double−𝑎𝑎rectangle

𝑀𝑀

isometric𝑎𝑎Γ%

Pillow box

Step1. We observe that, for any pillow box 𝑀𝑀, the base curves ofthe four cylinders of which 𝑀𝑀 consists are all congruent. We denote
one of them by Γ#:𝑧𝑧 = 𝑓𝑓 𝑥𝑥 (−𝑐𝑐 ≤ 𝑥𝑥 ≤ 𝑐𝑐) (see the picture below),
and it is sufficient to study only 1/4 of the pillow box.
Φ$:fold
along Γ4bend
(𝑥𝑥, 0, 𝑓𝑓 𝑥𝑥 )A rectangle(𝑥𝑥, 𝑓𝑓 𝑥𝑥 , 𝑓𝑓 𝑥𝑥 )

Γ%

1/4 of a pillow box．

Therefore, our problem becomes a problem for plane curves!

7

Outline of the proof of Theorem 1 (2) --- Step 2--Step2. We consider the following variational problem for plane curves.
For a given surface area, we maximize the enclosed volume of thepillow box given by a plane curve Γ#:𝑧𝑧 = 𝑓𝑓 𝑥𝑥 . Using the method of
Lagrange multiplier, we derive the Euler-Lagrange equation for Γ#:𝑧𝑧 =𝑓𝑓 𝑥𝑥 which gives a critical point of the functional “Area +𝜇𝜇・Volume”.
The result is the following ODE:!

$:

(1 + (𝑓𝑓 8 )$)9" 𝑓𝑓 88 = 1 𝑓𝑓 − 𝜇𝜇.
⋯ 1This equation means that the curvature 𝜅𝜅 of Γ# is a linear function of
the height, which implies that Γ# is an elastic curve.
bend

Φ$:fold

along Γ4

A rectangle

1/4 of a pillow box． 8

Outline of the proof of Theorem 1 (3) --- Step 3, 4, 5--Step3. We derive the boundary condition for our ODE:!

$:

(1 + (𝑓𝑓 8 )$)9" 𝑓𝑓 88 = 1 𝑓𝑓 − 𝜇𝜇.
⋯ 1in order that the solution gives a (local) maximum of volume. The
result is: the curve Γ#:𝑧𝑧 = 𝑓𝑓 𝑥𝑥 must be orthogonal to the 𝑥𝑥𝑥𝑥 plane.

Step4. We solve our ODE (1) for the curve Γ#:𝑧𝑧 = 𝑓𝑓 𝑥𝑥 under theboundary condition that the curve Γ# is orthogonal to the 𝑥𝑥𝑥𝑥 plane.

Step5 (final step). We prove the existence of the (global) maximum of
the volume of pillow boxes for any given double rectangle 𝑅𝑅 2𝑎𝑎, 2𝑏𝑏 .
bend

Φ$:fold

along Γ4

a rectangle

256

1/4 of a pillow box． 9

Representation of the optimal pillow box (I) --- base curves--The base curve Γ! :𝑧𝑧 = 𝑓𝑓 𝑥𝑥 of the optimal pillow box is represented as follows.
𝑥𝑥 = −𝐼𝐼" 𝑧𝑧 + 𝑐𝑐, 0 ≤ 𝑧𝑧 ≤ 𝑧𝑧! ,&𝑥𝑥 = 𝐼𝐼" 𝑧𝑧 − 𝑐𝑐, 0 ≤ 𝑧𝑧 ≤ 𝑧𝑧! ,#where，𝐼𝐼" 𝑧𝑧 ≔ ∫!

!

$"% &$"!
&$ "% &$"

0 ≤ 𝑥𝑥 ≤ 𝑐𝑐

⋯ 2−𝑐𝑐 ≤ 𝑥𝑥 ≤ 0#𝑑𝑑𝜁𝜁 > 0, (0 < 𝑧𝑧 < 𝑏𝑏), 𝑧𝑧! ≔

'
(

1− 1−

)
'"

,

𝑐𝑐: = 𝐼𝐼" 𝑧𝑧! ．𝜇𝜇 (< 0) is the curvature of 𝛤𝛤! at the end points that is determined by
the following.
#*%⋯ 3
𝑎𝑎 = ∫! $!

&$ "% &$"

bend

𝑧𝑧!

A rectangle#𝑆𝑆Φ$:fold

along Γ4𝑆𝑆&𝑆𝑆$

1/4 of a pillow box．

Representation of the optimal pillow box (II)--- surface and volume --Let Γ#:𝑧𝑧 = 𝑓𝑓 𝑥𝑥 be the base curve of the optimal pillow box givenin the previous slide.

The parts 𝑆𝑆4, 𝑆𝑆$ of the ¼ of the optimal pillow box are represented as
𝑆𝑆4 = { 𝑥𝑥, 𝑓𝑓 𝑥𝑥 , 𝑧𝑧 ; −𝑐𝑐 ≤ 𝑥𝑥 ≤ 𝑐𝑐, 0 ≤ 𝑧𝑧 ≤ 𝑓𝑓 𝑥𝑥 }J
⋯ 4
𝑆𝑆$ = { 𝑥𝑥, 𝑦𝑦, 𝑓𝑓 𝑥𝑥 ; −𝑐𝑐 ≤ 𝑥𝑥 ≤ 𝑐𝑐, 𝑓𝑓(𝑥𝑥) ≤ 𝑦𝑦 ≤ 𝑏𝑏}
Hence, the volume 𝑉𝑉(𝑓𝑓) of the optimal pillow box is;

𝑉𝑉 𝑓𝑓 = 4 F 𝑓𝑓 𝑥𝑥 𝑏𝑏 − 𝑓𝑓 𝑥𝑥
9;

bend

𝑧𝑧!

𝑆𝑆

A rectangle

𝑑𝑑𝑑𝑑

⋯ 5

Φ$:fold

along Γ4𝑆𝑆&𝑆𝑆$

1/4 of a pillow box．
11

Continuous isometric (i.e. not expanding,not contracting)deformation from a planar double rectangle to a pillow box
For application, it is important to obtain the explicit
representation from a planar region to a developable surface.
We can deform the initial double rectangle 𝑅𝑅 to any given pillow
box 𝑀𝑀 which is isometric to 𝑅𝑅 continuously and isometrically.
However, the crease pattern (the red curves in the pictures below)
is changed, which is not good for application.

𝑅𝑅

𝑀𝑀
An isometric deformation from 𝑅𝑅 to 𝑀𝑀
257

12

Isometric deformation from the single rectangle to 1/2of the pillow box without changing crease pattern! (I)The crease Γ4 of a pillow box is represented as 𝜂𝜂 𝑠𝑠 , 𝜁𝜁 𝑠𝑠 , 𝜁𝜁 𝑠𝑠 ,0 ≤ 𝑠𝑠 ≤ 𝐿𝐿 , where 𝑠𝑠 is arc-length parameter of Γ4 . Set
𝜑𝜑< 𝑠𝑠 = F=#

1 − 1 + 𝑡𝑡 $ 𝜁𝜁 8 𝑠𝑠$

𝑑𝑑𝑑𝑑 − 𝑐𝑐,

0 ≤ 𝑡𝑡 ≤ 1,
𝐶𝐶< 𝑠𝑠 = 𝜑𝜑< 𝑠𝑠 , 𝜁𝜁 𝑠𝑠 , 𝑡𝑡𝑡𝑡 𝑠𝑠 ,𝑞𝑞< 𝑠𝑠, 𝜏𝜏 = 𝐶𝐶< 𝑠𝑠 + 𝜏𝜏 W (0, 1, 0),
0 ≤ 𝜏𝜏 ≤ 𝑏𝑏 − 𝜁𝜁(𝑠𝑠).

Then, 𝐶𝐶A =𝛾𝛾B , 𝐶𝐶B =ΓB , and 𝑞𝑞C gives an isometricdeformation from ΩD to 𝑆𝑆D .
Ω$

𝑆𝑆&

𝛾𝛾&

𝑆𝑆$

13

Isometric deformation from the single rectangle to 1/2of the pillow box without changing crease pattern! (II)Next, set 𝑝𝑝! 𝑠𝑠, 𝜏𝜏 = 𝐶𝐶! 𝑠𝑠 − 𝜏𝜏𝛽𝛽! 𝑠𝑠 ,0 ≤ 𝑡𝑡 ≤ 1, 0 ≤ 𝑠𝑠 ≤ 𝐿𝐿, − 𝜁𝜁 𝑠𝑠 ≤ 𝜏𝜏 ≤ 0,

! !"#

"%!

Where 𝐶𝐶! 𝑠𝑠 = 𝜑𝜑! 𝑠𝑠 , 𝜁𝜁 𝑠𝑠 , 𝑡𝑡𝑡𝑡 𝑠𝑠 , 𝛽𝛽! 𝑠𝑠 = 0, ! !$# , ! !$# .

Then, 𝑝𝑝C gives an isometric deformation from ΩB to 𝑆𝑆B .
𝑝𝑝C with 𝑞𝑞C (in the previouspage) gives an isometric𝑆𝑆$
deformation from a rectangle Ω$
𝑆𝑆4 Γ4
to 1/4 of the pillow box.
𝛾𝛾
Ω #
4

By extending the above deformation using the reflection
with respect to the plane 𝑦𝑦 = 𝑏𝑏 , we obtain an isometric14
deformation from a single rectangle to 1/2 of the pillow box.

Future works
 For application, it is important to discuss “good”
discretization of surfaces with curved folds.
 Discuss continuous isometric deformations from ageneral developable surface with curved folds toplanar regions.

15

258

Summary We gave the definition of developable surfaces.
 We gave the existence, uniqueness, and representation formulaof the optimal pillow box.
 We gave a continuous isometric deformation (concretely) from aplanar region to a pillow box.
 We mentioned an application to architecture and discretizationin the talk in the workshop. Because this work is in progress, its
details are not included in this article.

16

259

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Learning Permutation Symmetry of a Gaussian
Vector
Bartosz Kolodziejek
Faculty of Mathematics and Information Science,Warsaw University of Technology, Poland
The study of hidden structures in data presents challenges in modern statistics andmachine learning. We introduce a Bayesian model selection approach, which allows to
identify permutation subgroup symmetries in Gaussian vectors. In other words, given a
finite iid sample of a p-dimensional Gaussian vector Z = (Z1 , . . . , Zp ) , we are looking
for a permutation subgroup Γ acting on {1, . . . , p} such that(Zi )pi=1 and (Zσ(i) )pi=1 have the same distributions

for any σ ∈ Γ. We also find the maximum likelihood estimate of the covariance matrix
in a Gaussian model obeying such symmetry restrictions. The talk is based on [1] and[2].
References[1] Graczyk, P., Ishi, H., Kolodziejek, B. and Massam, H. (2022) Model selection in the space ofGaussian models invariant by symmetry. Ann. Statist. 50, no. 3, pp. 1747-1774.
[2] Graczyk, P., Ishi, H. and Kolodziejek, B. (2022) Graphical Gaussian models associated to ahomogeneous graph with permutation symmetries, Physical Sciences Forum, 5(1), 20, pp. 1-9.

261

262

263

264

265

266

267

268

269

270

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Supercoiled structure of DNA and hyperelliptic
functionsShigeki Matsutani
Institute of Science and Engineering, Kanazawa University, JapanThe geometry of DNA has a helical structure as well as a more global supercoiled
structure. The geometry of this supercoiled structure is dominated by weak elastic
forces, but its geometry has not yet been mathematically described. Geometric models
that minimize its elastic energy, known as elasticae (elastic curves), cannot describe
the shape of DNA, even if three-dimensional effects are considered. Since 1997, thespeaker has been working to mathematically represent this shape by considering finite
temperature effects [1]. It is known from elementary considerations that the shape ofelastic curves under a finite temperature can be described by the hyperelliptic solutionof the modified KdV equation, which is a nonlinear integrable equation, in the twodimensional plane, and of the nonlinear Schrodinger equation in the three-dimensional
space. However, Abelian function theory, including hyperelliptic function theory, had
not reached the level where hyperelliptic function solutions could be specifically described and concretely treated at all as of 1997. Therefore, the speaker, together withlate Emma Previato since 2003, has restructured the Abelian function theory to thelevel of elliptic function theory, and has also developed related theories [2]. With Previato, he obtained certain shapes in 2022, albeit incomplete [3]. Although incompletemeans that it does not fully satisfy the reality condition, we were able to produce
mathematically shapes that have some features of the supercoiled structure of DNA,
albeit tentatively. This talk will describe the results obtained in 2022 and the processthat led to them.
The speaker has been studied novel devies and materials mathematically in researchand development of devices and materials for 27 years in Canon Inc. The usefulnessof mathematics, including the theory of singularity, in modern society will be briefly
discussed.
References[1] S. Matsutani, Statistical mechanics of elastica on a plane: origin of the MKdV hierarchy ,J. Phys. A: Math. & Gen., 31 (1998) 2705-2725.
[2] S. Matsutani, E. Previato, The Weierstrass sigma function in higher genus and applications tointegrable equations, (in preparation).
[3] S. Matsutani, E. Previato, An algebro-geometric model for the shape of supercoiled DNA Physica
D 430 (2022) 133073

271

Supercoiled structure of DNA andhyperelliptic functions
WORKSHOP on Mathematics for Industry 2023
September 28, (Thursday) 2023

Shigeki Matsutani
Kanazawa University

Menu
1. Self Introduction
2. Continuation of Self-Introduction
3. Supercoiled structure of DNA
4. Elastic curves
5. Statistical Mechanics of Elastic Curves
6. Excited states of elastic curves and the
MKdV equation
7. MKdV hyperelliptic curve solution for
genus 2
7.1 Review of the case of genus 1
7.2 Review of the case of genus 2
8. Future task

We live in an age in which we can create a
new reality by translating the real world into
mathematical language and investigating it.

Real World

Point Process
Quasi-conf
Conformal

Math World

How to do this?
Y.Shi, et.al, Physics Procedia 32 ( 2012 ) 389-394

272

Electron emission devices
1986-２０１０

Electron emission devices
M. Okuda, S. Matsutani, A.
Asai, A. Yamano, K. Hatanaka,
T. Hara, and T. Nakagiri

Caustics（singularity）
Orbits by
Numerical
Comp.

experiment

Production
Process ofeach emittercomputation

Electron emission devices
Conductive
particles

100μm
Energization
Change overtime

A Crack
appears

Y.Shi, et.al, Physics Procedia 32 ( 2012 ) 389-394

273

Electron emission devices

１[V]１．Scatter the conductive particles
as a Monte-Carlo computation.

0[V]

２．Solve the Dirichlet-Neumann
problem,div(γ・grad u)=0
numerically.

●conductive particle□high resistance matter

Computational Fluid Dynamics

We live in an age in which we can create a
new reality by translating the real world into
mathematical language and investigating it.

Real World

Stratification
Inf. Lie. Alg.
Phase field

Math World{0} ⊂V1⊂ V2

浅井朗 ながれ 24（2005)

How to do this?

274Computational Fluid Dynamics
Modeling the fluid in the discharge section of theinkjet printer2004 M Japan Patent2006-30060
2011 M-Nakano-Shinjo
discharge section
Ink-drop

Triple phase interface is singular
Triple phases: solid, liquid and gas.
How to model the triple junction?

{0} ⊂V1⊂ V2

gas-drop heater
Watanabe-Shinjo：
日本機械学会誌 2012 We regard it as stratification for the modeling

Computational Fluid Dynamics
Modeling the fluid in the discharge section of theinkjet printer2004 M Japan Patent2006-30060
2011 M-Nakano-Shinjo
discharge section
gas-drop
Ink-drop

Triplephase interface is singular
Ink-drop
Triple phases: solid, liquid and gas.
How to model the triple junction?

{0} ⊂V1⊂ V2

gas-drop heater
Watanabe-Shinjo：
日本機械学会誌 2012 We regard it as stratification for the modeling

Computational Fluid Dynamics
Observation

Modeling the fluid in the discharge section of theinkjet printer2004 M Japan Patent2006-30060
2011 M-Nakano-Shinjo
discharge section
Computation

Triple phase interface is singular
Triple phases: solid, liquid and gas.
How to model the triple junction?

{0} ⊂V1⊂ V2

heaterWe regard it as stratification for the modeling
Asai, ながれ 24（2005)

275

Nano-matrials

We live in an age in which we can create a
new reality by translating the real world into
mathematical language and investigating it.

Real World

Point Process
Quasi-conf
Conformal

Math World

How to do this?
Preparation and Some Properties of a Nanocomposite ofPolyacrylonitrile with Acetylene Black Arjun Maity and Mukul Biswas
Polymer Journal, Vol. 36 (2004) No. 10
pp.812-816

Nano-matrials
Material design of key components of LBPs
Mathematical modeling using percolation theory.
2015M-Shimosako

How to control the highresistivity by mixing
conductive carbon
nanoparticles with a highresistivity polymer matrix
to preserve its property
for a long time.

Preparation and Some Properties of a Nanocomposite ofPolyacrylonitrile with Acetylene Black Arjun Maity and Mukul Biswas
Polymer Journal, Vol. 36 (2004) No. 10
pp.812-816

276

Nano-matrials
Material design of key components of LBPs
Mathematical modeling using percolation theory.
2015M-Shimosako

3D potential distribution
Computational simulation

3D potential distribution
Computational simulation

We live in an age in which we can create a
new reality by translating the real world into
mathematical language and investigating it.

Real World

Point Process
Quasi-conf
Conformal

Math World

Perspectives from industry,
1. Cutting-edge technology
requires cutting-edge mathematics.
2. Some of the mathematics fortechnology is so profound.
How to do this?

Menu
1. Self Introduction
2. Continuation of Self-Introduction
3. Supercoiled structure of DNA
4. Elastic curves
5. Statistical Mechanics of Elastic Curves
6. Excited states of elastic curves and the
MKdV equation
7. MKdV hyperelliptic curve solution for
genus 2
7.1 Review of the case of genus 1
7.2 Review of the case of genus 2
8. Future task

277

Supercoil in DNA

Electron microscope image of DNA

Supercoil in DNA
DNA forms
・double coil structure and・supercoil structure.

Supercoil structure is・governed by weak elastic
forces.
・But not an elastic curve

Menu
1. Self Introduction
2. Continuation of Self-Introduction
3. Supercoiled structure of DNA
4. Elastic curves
5. Statistical Mechanics of Elastic Curves
6. Excited states of elastic curves and the
MKdV equation
7. MKdV hyperelliptic curve solution for
genus 2
7.1 Review of the case of genus 1
7.2 Review of the case of genus 2
8. Future task

278

What is an elastica (elastic curve)?
Elastica is a curve on a (complex)
plane determined by elastic forces.

C
ϕ

n

Ｙ
Ｘ

What is an elastica (elastic curve)?
Z : N → C :analytic immersion(|∂s Z| = 1).

N=S 1 or =[0, 1]

C

s ：arclengthϕZ(s )= X( s )+ iY(s ),t = ∂s Z = eiϕ,nＹ(Φ ∈ Cω (N, R))= cos ϕ + i sin ϕＸk :=∂s ϕ : curvature; k = 1/[curvature radius].

What is an elastica (elastic curve)?
Curvature& Frenet-Serret relationt :=∂s Z :tangential vector, n:=i i∂s Z:normal vector∂s t = k n,

∂s n = −k t, (∂22s Z = ik∂s Z )k := ∂s ϕ : curvature; k = 1/[curvature radius].

279

What is an elastica (elastic curve)?
Curvature& Frenet-Serret relationt :=∂s Z :tangential vector, n:=i i∂s Z:normal vector∂s n = −k t, (∂22s Z = ik∂s Z )∂s t = k n,

k := ∂s ϕ : curvature; k = 1/[curvature radius].
Elastica Problem (Jacob/Daniel Bernoulli-Euler (1691-1744))

Determine the shape of the elastic curve thatexists on the plane mathematically!

⇔ Find the shape that minimizes the energy
under the iso-arc length.

Infinitesimal isometric deformation
Infinitesimalvariation::

Infinitesimal isometric deformation

Static modified KdV equation

Elastica
to findthe shape of theStaticProblem
modifiedisKdV
equationcurve whose curvature obeys the static
modified KdV equation
It is a prototype of the nonlinear
integrable system.

280

Infinitesimal isometric deformation

Infinitesimal isometric deformation

Elliptic curve

Elliptic curve

281

Elliptic curve

y
e1

e2 e3

∞

x

x,y are regarded as complex numbersx→ x +x’√-1, y→ y +y’√-1

Meromorphic function
over elliptic curve＝Elliptic function
Elastica is expressed by the elliptic functions!

Elastica (Elastic curve):
Euler’s solutions

It is a solution of MKdV equationfrom a modern point of view.

282

Elastica (Elastic curve):

Real transcendental curves

Euler’s sketch by numerical comp.(1744)

Menu
1. Self Introduction
2. Continuation of Self-Introduction
3. Supercoiled structure of DNA
4. Elastic curves
5. Statistical Mechanics of Elastic Curves
6. Excited states of elastic curves and the
MKdV equation
7. MKdV hyperelliptic curve solution for
genus 2
7.1 Review of the case of genus 1
7.2 Review of the case of genus 2
8. Future task

Statistical mechanics of elastica:
The geometric structure of the parameter
space (moduli) of a shape is unknown:

Find orbits with iso-energy.
E.Previato2015 SM 1997

283

Menu
1. Self Introduction
2. Continuation of Self-Introduction
3. Supercoiled structure of DNA
4. Elastic curves
5. Statistical Mechanics of Elastic Curves
6. Excited states of elastic curves and the
MKdV equation
7. MKdV hyperelliptic curve solution for
genus 2
7.1 Review of the case of genus 1
7.2 Review of the case of genus 2
8. Future task

Formulation of iso-energetic geometry

Analytic, isometric

Find orbits with the iso-energy.
E.Previato2015 SM 1997

MKdV equation

Solutions of the MKdV equation preserve
for the time-development

The time t is not physical time but a parameterin the moduli of the immersion of the curve284

MKdV equation

MKdV equation contains the static MKdV
equation of elastica as t=s.
⇔ It is a natural generalization of elastica

Statistical mechanics of elastica:
Find orbits with iso-energy
Find higher-order solutions of the
MKdV equation!
E.Previato2015 SM 1997

MKdV equation

1. The MKdV equation has hyperelliptic
function solutions.
2. Due to the higher genus of hyperelliptic
curves (compact Riemann surfaces), thesolutions are expected to express more
complicated (elastic) curves.

285

Elastic energy

5.70
4.00
excited hyperelliptic solutions ta teof MKdV equation

1.42
1.00
0.00

gr ou nds ta teelliptic solution of
static MKdV equation

statesStatistical mechanics of elastica
Assign the appropriate topology in the parameter spaceof the geometry (moduli),formulate the above integral
in terms of the measures determined from theBoltzmann weights of the Euler-Bernoulli energyfunctional, and perform the integration.

1. Construct hyperelliptic solutions to
the MKdV equation of higher genus.
2. Extract "real" part of hyperelliptic
Jacobi variety as the moduli of"real" hyperelliptic curves over C.
M 1997, M-Onishi 2001, M-Previato 2015

Statistical mechanics of elastica
Assign the appropriate topology in the parameter spaceof the geometry (moduli),formulate the above integral
in terms of the measures determined from the
1. To Construct solutions to theBoltzmann weights of the Euler-Bernoulli energy
equationfunctional, MKdVand performthe integration.

2. To
find real part"of hyperelliptic
If hyperelliptic
function
theory had thesame levelof sophisticationand ofJacobivarietyas the moduli
concretenessas Weierstrass’
elliptic"real"
hyperelliptic
curves.
function theory, this problem would be
solved! but it is not at that level.

Reconstruct the theory of hyperelliptic(Abelian) functions to have the same level
as the theory of elliptic functions.
M The
1997,
M-Onishi
M-Previato 2015
main
theme2001,
is a reconstructionof
Abelian function theory with E.Previato

286

Menu
1. Self Introduction
2. Continuation of Self-Introduction
3. Supercoiled structure of DNA
4. Elastic curves
5. Statistical Mechanics of Elastic Curves
6. Excited states of elastic curves and the
MKdV equation
7. MKdV hyperelliptic curve solution for
genus 2
7.1 Review of the case of genus 1
7.2 Review of the case of genus 2
8. Future task

MKdV solutions of genus two
・Review the genus one case・Step to genus twoS.M., and Emma Previato,

An algebro-geometric model for the
shape of supercoiled DNA
Physica D, 2022

Review the genus one case:

e2

S1
e1

They satisfy the SMKdV eq.

e3

287

Review the genus one case:
Meromorphic function
over elliptic curve＝Elliptic function
Elastica is expressed by the elliptic functions!

Review the genus one case:

SMKdV eqを満たす

e2

S1
e1

e3

Review the genus one case:

Num.Comp.

e2

S1
e1

Euler
methode3288

a=1.2a=1.25

Closed elasticae

g=0

g=1

MKdV solutions of genus two
・Review the genus one case・Step to genus twoS.M., and Emma Previato,

An algebro-geometric model for the
shape of supercoiled DNA
Physica D, 2022

289

Genus1 y
e1

e2 e3

Genus2y∞∞b1 b2 b3 b4 b5x

x

Genus two case:

S1
b1

b2

b4

b3

b5

Genus two case:

S1
b1

b2

b4

b3

b5
290

Genus two case:

ＭKdV equation/C
b2 b4
S1
b1

b3

b5

Genus two case:２

Meromorphic function overhyperelliptic curve＝Hyperelliptic function
Shape of elastica is determined byhyperelliptic functions

It is a direct
generalisation.

291

Genus two case:
ＭKdV equation/C

ＭKdV equation with gauge field/Ｒ

Genus two case:
ＭKdV equation with gauge field/Ｒ

ＭKdV equation/Ｒ

Genus two case:
ＭKdV equation with gauge field/Ｒ

ＭKdV equation/Ｒ
4=Dim. of parameter spaceof hyperelliptc curve g=2

4 conditions

292

Dim of sol. ofMKdV(t&s)=2

Genus two case:
ＭKdV equation with gauge field/Ｒ

ＭKdV equation/Ｒ
4=Dim. of parameter spaceof hyperelliptc curve g=2

4 ill-posed
conditions

Dim of sol. ofMKdV(t&s)=2

Genus two case:
ＭKdV equation/Ｒ
4 conditions

4=Dim. of parameter spaceof hyperelliptc curve g=2

ill-posed

Dim of sol. ofMKdV(t&s)=2

Need to adjust the parameters (bi) of thecurve itself to find the situation where theconditions are degenerate⇒ Extremely difficult(only possible with more than genus 3?)
First, loosen the conditions,
Investigate the properties of g = 2

Genus two case:
ＭKdV equation/Ｒ
4 conditions
4=Dim. of parameter space
Dim of sol. of
concludecurvethatg=2
in this stage, the MKdV(t&s)=2
ofWe
hyperelliptc
ill-posed
hyperelliptic curves X withgenus two cannot
exhibit the generalized elastica well because
Needto adjusttheparametersbi)bothof thewecannotextract
thereal parts (inX
curve
itselftofind
thesituationwhere
theand its Jacobi variety JX over C. S.M., andconditions
are ,E.
degenerate(S.M.
Previato, Physica D, 2022)⇒ Extremely difficult(only possible with more than genus 3?)
First, loosen the conditions,
Investigate the properties of g = 2
293

Genus two case:
ＭKdV equation/C
3 conditions
a=1.2

Genus two case:

S1
b1

b2

b4

b3

b5

Genus two case:

S1
b1

b2

b4

b3

b5
294

Genus two case:

S1
b1

b2

b4

b3

b5

Genus two case:
Num.Comp.

S1
b1

b2

bb44

b3

b5

Euler
methodGenus two case:
(a)
(b)
(c)
(d)
(e)
(f)
They are solutions of MKdV
equation over C but not over R295

M. Previato
Physica D, 2022

Genus two case:
The orbit s on the complex planeCs

M. Previato
Physica D, 2022

g=2
Electron Micrographs of DNA

S.Matsutani E.Previato, 2021

Fundamentals of Biochemistry: Life at the Molecular
Level, 3rd Ed. D. Voet, J. Voet, C. Pratt

Future tasks:
1.
2.
3.

to investigate the cases of g > 3.
to evaluate the moduli space ofgeneralized elastica analytically andnumericallyto extend them to a generalized
elasitca in R3 by finding thehyperelliptic solutions of non-linear
Schrodinger equation in our novel
approach with Emma Previato.

296

Thank you!

297

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Information geometry of positive measures
Naomichi Nakajima
Faculty of Science and Engineering, Waseda University, Japan
Information geometry brings a united geometric insight into various aspects of statistical science, machine learning and so on by regarding the parameter space of a statisticalmodel as a Riemannian manifold equipped with the Fisher-Rao metric. The dually flat
structure on a Riemannian manifold introduced by Amari-Nagaoka takes a central rolein information geometry. It is known that the space of probability distributions on a finite set naturally has the dually flat structure. For this space, Amari has characterized
the dually flat structure from the viewpoint of statistics through defining the space of
positive measures simply by removing the normalization condition. On the other hand,
we have developed the counterpart for the space of transition probabilities of a given
Markov chain, which may provide a new geometric insight into Markov chains. In thispresentation, I would like to talk about Amari’s theory and our theory for Markov
chains.

299

Information geometry of positive measures
Naomichi Nakajima
Waseda University, Japan
Sep. 29th, 2023
WORKSHOP on Mathematics for Industry@Warsaw University of Technology
supp.: JSPS KAKENHI Grant Number 22KJ0052, 22KK0034
1 / 23

Summary of my talk• Information geometry brings a united geometric insight on various fields such as
statistics, machine learning, optimization theory and so on. In information geometry,a statistical model is regarded as a Riemannian manifold endowed with thecertainduality,
Fisher-Rao metric and two kinds of affine connections satisfying acalled a statistical manifold.
• A dually flat manifold is a statistical mfd with flat connections, that takes acentral role in information geometry, introduced by Amari-Nagaoka.
• Regarding dually flat structures, there is a well established theory of positive
measures on a finite set S due to Amari. It investigates dually flat structures of thespace P(S) of probability distributions on S in terms of some “asymmetric distancefunction” on P(S), called a divergence.
2 / 23

Summary of my talk
• On the other hand, information geometry of Markov chains has been studied by
Nagaoka and others using the dually flat structure of the space of transition
probabilities.
• In comparison with information geometry of P(S), roughly speaking, the studies
above are on information geometry of the space of conditional probabilities.
▶ Main topic of my talk.
We will investigate the counterpart for a Markov chain of Amari’s theory of positive
measures. This study does not only investigate information geometry of the specificmodel, a Markov chain, but also suggests a new direction of statistics of conditional
probabilities.
3 / 23300

Contents• Backgrounds• Statistical manifolds, dually flat manifolds and divergences• Amari’s theory of positive measures on a finite set [Amari]• Information geometry of transition probabilities of a given Markov chain [Nagaoka]• Our theory for transition probabilities [N][N] The space of positive transition measures of a Markov model, in preparation.
[Amari] S. Amari, α-divergence is unique, belonging to both f -divergence and Bregman divergenceclasses, IEEE Trans. Inform. Theory 55 (2009), 4925–4931.
[Nagaoka] H. Nagaoka, The exponential family of Markov chains and its information geometry,
Proceedings of The 28th Symposium on Information Theory and Its Applications(SITA2005) (2005).
4 / 23

Statistical manifolds, dually flat manifolds and divergences

Let (M, h) be a pseudo-Riem. mfd and ∇ a torsion-free affine connection of T M .
• The triplet (M, h, ∇) is a statistical mfd if the cubic tensor C := ∇h istotally symmetric. Then C is called the Amari-Chentsov tensor [3, 4].

• The dual connection ∇∗ of ∇ w.r.t. h is defined byXh(Y, Z) = h(∇X Y, Z) + h(Y, ∇∗X Z) (X, Y, Z ∈ X(M ))

Also, an “asymmetric distance” ρ : M × M → R induces (h, ∇, ∇∗ ) on M as follows.
For vector fields X1 , · · · , Xk , Y1 , · · · , Yl on M , define the function
ρ[X1 · · · Xk |Y1 · · · Yl ] : M → R,

ρ[X1 · · · Xk |Y1 · · · Yl ](r) = (X1 )p · · · (Xk )p (Y1 )q · · · (Yl )q (ρ(p, q))|p=q=r .
5 / 23

Statistical manifolds, dually flat manifolds and divergences
We call ρ a contrast function if it satisfies (i) ρ[−|−](r) = ρ(r, r) = 0,(ii) ρ[X|−](r) = ρ[−|X](r) = 0,
 (iii) −ρ[X|Y ] : pseudo-Riemannian metric on M.
We call ρ a weak contrast function if it satisfies only (i) and (ii) ([N.-Ohmoto2021]).
[Eguchi1992]
Contrast
Functionsmulti : 1

Statistical
Manifolds[Matumoto1993]
restores301

6 / 23

Statistical manifolds, dually flat manifolds and divergences
For a statistical mfd (M, h, ∇), ∇ is flat ⇐⇒ its dual connection ∇∗ is flat.
Definition (Amari-Nagaoka [3, 4])A statistical mfd (M, h, ∇, ∇∗ ) is a dually flat mfd if ∇ is flat. Then we also call(h, ∇, ∇∗ ) the dually flat structure of M .
We write θ = (θ1 , · · · , θn ) for ∇-affine coords. Put ∂i := ∂θ∂ i . Then there exists a
potential function f (θ) on θ s.t.
1. the metric h is locally given by the Hessian matrix of f (θ): h(∂i , ∂j ) = ∂i ∂j f ,∂f) gives ∇∗ -affine coordinates, called
2. the gradient map η = (η1 , · · · , ηn ) (ηi := ∂θithe dual coordinates of θ,
Another definition (Hessian structure [Shima]):Given a (M, h, (∇, θ), f (θ)) with h = ∂i ∂j f
∂f
 define the dual flat connection and the dual coord (∇∗ , η = (ηi )) by ηi := ∂θi

7 / 23

Statistical manifolds, dually flat manifolds and divergences
A dually flat mfd (M, h, ∇, ∇∗ ) has the canonical contrast function D : M × M → R,called the Bregman divergence:∂f(θ(q))T (θ(q) − θ(p)) (p, q ∈ M ),D(p, q) = f (θ(p)) − f (θ(q)) +∂θwhere f (θ) is a potential function of M . (strictly speaking, D is defined on an open
neighborhood of the diagonal set of M )Remark:• The definition of D is independent of the choice of (θ, f (θ)).

• D restores the dually flat structure (h, ∇, ∇∗ ), i.e.,h(X, Y ) = D[X|Y ],h(∇X Y, Z) = −D[XY |Z], h(∇∗X Y, Z) = −D[Z|XY ].
8 / 23

Example: the space of discrete distributions• S = {0, 1, · · · , n}: a finite set
n• P(S) = {(p0 , p1 , · · · , pn ) ∈ Rn+1 | pi > 0 and
i=0 pi = 1}

We call P(S) the space of discrete distributions on S. Take a system of coordinates(p1 , · · · , pn ) (p0 = 1 − p1 − · · · pn ). We regard it as flat coordinates (∇, η = (ηi )ni=1 ):ηi := pi (the expectation parameters).

Thenϕ(η) =

ni=1 pi log pi

is a convex function, known as the negative entropy in statistics.
Hence the metric h is defined by2ϕh( ∂η∂ i , ∂η∂ j ) = ∂η∂i ∂η
.
jTherefore, (P(S), h, (∇, η), ϕ(η)) is a dually flat mfd (Hessian mfd).

302

9 / 23

Example: the space of discrete distributions
Importantly, the Bregman divergence D : P(S) × P(S) → R induced by ϕ is the
KL-divergence on P(S), i.e.,npi
pi log =: KL[p, q],D(p, q) =qi
i=0

where p = (p0 , · · · , pn ), q = (q0 , · · · , qn ) ∈ P(S).

• We consider the following problem: are there any other contrast functions to derivea dually flat structure of P(S)?

• Of course, for example, we consider a quadratic function as a potential function, andthen it derives another dually flat structure of P(S).
• We are interested in the dually flat structure with “statistical invariance”, which is acertain condition required from statistics.

10 / 23

The space of positive measures on a finite set• Amari has introduced the space P̄(S) of positive measures on S as an extended
space of P(S) and investigated the problem above by finding the Bregman and
F -divergence on P̄(S) suitably.

• An F -divergence DF on P̄(S) is a contrast function, and it is known that thestatistical manifold structure induced by DF of P̄(S) satisfies statistical invariance.
• In [Amari], Amari has shown that the KL-divergence DKL on P̄(S) is the only
contrast function such that– it is both a Bregman divergence and an F -divergence,– it and its restriction to P(S) induce the dually flat structures of P̄(S) and P(S),respectively.

11 / 23

The space of positive measures on a finite set and F -divergences• S = {0, 1, · · · , n}: a finite set• P̄(S) = {(p0 , p1 , · · · , pn ) ∈ Rn+1 | pi > 0} ⊃ P(S) = {pi > 0 andWe call P̄(S) the space of positive measures on S.
Given a strictly convex function F : (0, ∞) → R withni=0 pi = 1}

F (1) = F ′ (1) = 0 and F ′′ (1) = 1,

called a standard convex function [Amari], the function DF : P̄(S) × P̄(S) → R
defined by
 
n

qi
pi F
DF (p, q) =pi
i=0
is called the F -divergence on P̄(S), where p = (p0 , · · · , pn ), q = (q0 , · · · , qn ).

12 / 23

303

The space of positive measures on a finite set and F -divergences
In the case where F (t) = − log t + (t − 1), the F -divergence DF is the KL-divergenceon P̄(S):
  
n
n
n


pi
+
pi logqi −pi .
DF (p, q) =
qi
i=0
i=0
i=0
• In fact, P̄(S) has the dually flat structure; its flat coordinates are η = (p0 , · · · , pn )and the potential function ϕ(η) is given byn
pi log pi .
ϕ(η) =• For p, q ∈ P(S), it holds thatni=0 pi =nDF (p, q) =

i=0

n

i=0 qi = 1, which yields

pi logi=0 pi= KL[p, q].
qi

13 / 23

Information geometry of the space of transition probabilities Setting:• X = {0, 1, · · · , d}: a finite set• E ⊂ X × X : a subset
 We regard (X , E) as a direct graph.
• F + = {f : E → R | f (x, y) > 0 for any (x, y) ∈ E}
• W = {w ∈ F + | y:(x,y)∈E w(x, y) = 1 for any x ∈ E} ⊂ F +We call w ∈ W a transition probability on E (the word “transition probability” comesfrom Markov chains).

14 / 23

Information geometry of the space of transition probabilities
We assume that E is strongly connected, that is, for any x, y ∈ X there exist(x1 , x2 ), (x2 , x3 ), · · · , (xN −1 , xN ) ∈ E such that x1 = x, xN = y (N ≥ 2).
By this assumption, for every f ∈ F + we can apply the Perron-Frobenius theorem tof (i, j) (i, j) ∈ E
A(f ) = [aij (f )]0≤i,j≤d , aij (f ) =0(i, j) ∈/E
Then we get a unique real value r(f ) > 0 and vector µf = (µf (0), · · · , µf (d))Tsatisfying• r(f ) is the Perron-Frobenius root, which is an eigenvalue of A(f ),• µf is a left eigenvector associated with r(f ) such that µf (i) > 0 for any i, anddi=0 µf (i) = 1.

We call the vector µf the stationary distribution for f .

15 / 23304

Information geometry of the space of transition probabilities
We consider the following two spaces:
M = {η = (ηxy )(x,y)∈E ∈ R|E| | ηxy > 0},ηxy = 1 and
M = {η = (ηxy )(x,y)∈E ∈ M |(x,y)∈E
y:(x,y)∈E ηxy =
y:(y,x)∈E ηyx for any x ∈ X }.

In [Nagaoka], it is shown that W is a dually flat manifold, and its expectation
parameter space is M .

Theorem ([Nagaoka])
1. The mapping T : W → M , w → (µw (x)w(x, y))(x,y)∈E is a diffeomorphism.
2. There exists a convex function ϕ : M → R; the Bregman divergence
D : W × W → R induced by ϕ is
w1 (x, y).
µw1 (x)w1 (x, y) log
D(w1 , w2 ) =w2 (x, y)(x,y)∈E

16 / 23

Positive transition measures on W (our work)▶ Aim: We construct the counterpart of Amari’s picture in (P(S), P̄(S)) for W.
▶ Main results:• We extend W to the bigger space F + .
• We define an F -divergence on F + and a diffeomorphism T̄ between F + and M .
• We give a divergence that is both a Bregman divergence and an F -divergence.
• Actually, the potential function ϕ̄ has a 1-dimensional kernel of its Hessian matrix atevery point of M , thus we take a hyperplane section M̃ in M so that a genuine duallyflat structure is defined on it. That induces a hypersurface W̃ in F + .

F+


W̃


W

∼

M

T̄

∼

T̄ |W̃
∼
T

ϕ̄



 M̃

M


ϕ̄|M̃

ϕ

R
R

17 / 23

Positive transition measures on W (our work)
Definition ([N])
Let F : (0, ∞) → R be a strictly convex function with F (1) = F  (1) = 0 and F  (1).
We define the F -divergence on F + as DF : F + × F + → R,f (x,y)DF (f, g) = (x,y)∈E µf (x)f (x, y)F g(x,y)
.
r(g)
r(f )
Proposition ([N])

The F -divergence DF has the following properties:
1. DF (f, g) ≥ 0.
2. DF (f, g) = 0 if and only if g = af for some a > 0.
3. DF is a weak contrast function on F + . Let hF denote the symmetric (0, 2)-tensor
on F + induced by DF .
4. The null space of hF at f ∈ F + is the tangent space of the halfline{af | a > 0} ⊂ F + .
18 / 23305

Positive transition measures on W (our work)
We setT̄ : F + → M , f → (µf (x)f (x, y))(x,y)∈E .
We also set for η = (ηxy )(x,y)∈E ∈ Mr(η) :=

ηxy .

(x,y)∈E

Lemma ([N])
T̄ has the following properties:∼
1. T̄ is a diffeomorphism, and T̄ |W = T : W → M .
2. T̄ (af ) = aT̄ (f ) for f ∈ F + and a > 0.
3. r(f ) = r(η) with T̄ (f ) = η.

19 / 23

Positive transition measures on W (our work)
Theorem ([N])
Let F (t) = − log t + (t − 1). Then the F -divergence is the Bregman divergencegiven by the following potential function on M :ϕ̄(η) =
ηxy log ηxy −ηx log η x .

(1)

x∈X

(x,y)∈E

For w1 , w2 ∈ W we see
2 (x,y)DF (w1 , w2 ) = (x,y)∈E µw1 (x)w1 (x, y)F w
w1 (x,y)
1 (x,y)= (x,y)∈E µw1 (x)w1 (x, y) log w+ (x,y)∈E µw1 (x)(w2 (x, y) − w1 (x, y))w2 (x,y)
1 (x,y)= (x,y)∈E µw1 (x)w1 (x, y) log w: the divergence by Nagaoka
w2 (x,y)20 / 23

Positive transition measures on W (our work)

We see that the Hessian matrix of ϕ̄ at every point η ∈ M has the 1-dimensional
kernel spanned by the numerical vector η ∈ R|E| ∼= Tη M .

Therefore, by imposing only the normalization condition (x,y)∈E ηxy = 1 on M , wehave the hyperplane section M̃ in M so that ϕ̄ is strictly convex on it:M̃ := {η = (ηxy ) ∈ M | r(η) = 1}.

Using the relation r(f ) = r(η) with T̄ (f ) = η , we get the genuine dually flat manifold
W̃ = {f ∈ F + | r(f ) = 1},

which is an extended space of W as a hypersurface in F + .

21 / 23306

Positive transition measures on W (our work)
Theorem ([N])
The hypersurface W̃ has the dually flat structure induced by the potential functionϕ̃ := ϕ̄|M̃ on M̃ ; the restriction of this dually flat structure to W restores the duallyflat structure of [Nagaoka]. We call W̃ the space of positive transition measures.
Moreover F -divergences on W̃ are written as
(f, g ∈ W̃).
DF (f, g) = (x,y)∈E µf (x)f (x, y)F fg(x,y)
(x,y)
F+


W̃


W

∼

M

T̄

∼

T̄ |W̃
∼
T

ϕ̄



 M̃

M


ϕ̄|M̃

ϕ

R
R

22 / 23

Summary and future plans• We have defined the class of F -divergences on F + and given a divergence which isboth a Bregman divergence and an F -divergence. Moreover, we have given a duallyflat manifold W̃ which is an extension of W by analyzing the kernels of the potentialfunction ϕ̄ on M .
• In order to completely establish the counterpart of Amari’s theory for the pair(W, W̃), we need some discussions from the view point of statistics.

• In the first place, the “statistical invariance” for conditional probabilities must be
discussed.
• Then, F -divergences should be characterized by the statistical invariance above.
• Besides, a divergence on W̃ which is both a Bregman divergence and an
F -divergence may be uniquely determined under certain conditions.
23 / 23

Thank you for your attention!

23 / 23307

Reference i[1] N. Nakajima and T. Ohmoto, The dually flat structure for singular models, Info.
Geom., 4 (2021), 31–64.
[2] S. Amari, Information geometry of the EM and em algorithms for neural networks,
Neural Networks, 8 (1995), 1379–1408.
[3] S. Amari, Information Geometry and Its Application, Applied Math. Sci., 194,
Springer(2016)[4] S. Amari and H. Nagaoka, Methods of Information Geometry, A.M.S., Oxford
Univ. Press (2000).
[5] S. Eguchi, Geometry of minimum contrast, Hiroshima Math. J., 22 (1992),
631–647.
23 / 23

Reference ii[6] T. Matumoto, Any statistical manifold has a contrast function – On the
C 3 -functions taking the minimum at the diagonal of the product manifold,
Hiroshima Math. J., 23 (1993), 327–332.
[7] H. Nagaoka, The exponential family of Markov chains and its informationgeometry, Proceedings of The 28th Symposium on Information Theory and Its
Applications (2005).
[8] H. Shima, The geometry of Hessian Structures, World Scientific (2007).
[9] J. Takeuchi, Fisher information determinant and stochastic complexity for Markov
models, 2009 IEEE International Symposium on Information Theory (2009),
1894–1898.

23 / 23308

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Multivariate Hawkes processes with graphs
Mariusz Nieweglowski
Faculty of Mathematics and Information Science,Warsaw University of Technology, Poland
A very interesting and important class of stochastic processes was introduced by Alan
Hawkes in [1]. These processes, called now Hawkes processes, are meant to model
self-exciting and mutually-exciting random phenomena that evolve in time. The selfexciting phenomena are modeled as univariate Hawkes processes, and the mutuallyexciting phenomena are modeled as multivariate Hawkes processes. The Hawkes processes have been applied to modeling in meany areas of science, including: insurance,
finance, seismology and neurology. In this talk we provide some results on markovianity of the Generalized Multivariate Hawkes Processes (GMHP) introduced in ourearlier papers. GMHP are multivariate marked point processes that add an importantfeature to the family of the (classical) multivariate Hawkes processes: they allow for explicit modelling of simultaneous occurrence of excitation events coming from different
sources, i.e. caused by different coordinates of the multivariate process. We proposethat this structure of mutual excitations is specified in terms of the excitation graph.
We provide results which show that under some conditions on its kernels the intensitiesof GMHP’s are functions of a Markov processes. Moreover we show that it is possible
to compute their Laplace transform by means of system of ODE’s. The talk is basedon [4].
References[1] A.G. Hawkes, ”Spectra of Some Self-Exciting and Mutually Exciting Point Processes”, Biometrika
58(1):83–90, 1971.
[2] T.R. Bielecki, J. Jakubowski, M. Nieweglowski, ”Construction and Simulation of Generalized
Multivariate Hawkes Processes”, Methodology and Computing in Applied Probability (2022)
24:2865-2896[3] T.R. Bielecki, J. Jakubowski, M. Nieweglowski, ”Multivariate Hawkes processes with simultaneous occurrence of excitation events coming from different sources”, Stochastic Models (2022) online[4] T.R. Bielecki, J. Jakubowski, M. Nieweglowski, ”Markovianization of Multivariate Hawkes
processes”, preprint

309

310

311

312

313

314

315

316

317

318

319

320

321

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

On envelopes created by circle families in the plane
Takashi Nishimura
Faculty of Environment and Information Sciences,
Yokohama National University, Japan(joint work with Yongqiao Wang)
Envelopes of planar curve families have fascinated many pioneers since the dawn ofdifferential analysis. In most typical cases, straight line families have been studied.
However, even for envelopes created by straight line falimies, to our surprize, there
were several unsolved problems until very recently. In my talk at WAAS, recently
discovered answers to these problems were explained.
On the other hand, circle families in the plane are non-negligible families because
the envelopes of them have already had important applications to Industry. In thistalk, firstly, as one of important applications of envelopes of circle families to Industry,
the so-called “Mohr failure envelope” is introduced. After that, a general theory forenvelopes of circle families shall be explained.

323

On envelopes created by circle families in theplane (a joint work with Yongqiao Wang)
Takashi Nishimura(Yokohama National University)

Reference[WN] Yongqiao Wang and T.N., Envelopes created bycircle families in the plane, preprint.
(available at https://arxiv.org/abs/2301.04478)

1§1. Soil Mechanics
Circle families in the plane are non-negligible families
because the envelopes of them have already had important applications. As one of application of circle
family in the plane, Let me ﬁrst explain the so-called
Mohr failure envelope in the ﬁeld “Soil Mechanics”.

2324

In analysis of the stability of soil masses, the shear
strength τf of a soil at a point on a particular plane
is expressed as a linear function of the eﬀective normal
stress σf at failure:τf = σf tan φ + c,
where φ and c are the angle of shearing resistance andcohesion intercept respectively. A method using Mohr
circles to obtain the shear strength parameters φ and ccan be found (for instance) in “R.F. Craig, Craig’s soil
mechanics, Seventh edition, Taylor and Francis Group
Press, New York, 2004. ISBN: 9780415332941”. A
brief description of this method is given as follows.
3

The stress state of a soil can be represented by a Mohr
circle which is deﬁned by the eﬀective principal stresses
σ1 and σ2. The center and the radii of the Mohr cirσ1 −σ22
cle are ( σ1+σ2 , 0) and2 , respectively. By experiments, one can obtain some values of eﬀective principal
stresses σ1 and σ2 at failure. The Mohr circles in terms
of eﬀective principal stress are drawn in Figure 1.

4

Figure 1

5

325

The envelope created by Mohr circles is called the Mohrfailure envelope which may be a slightly curved curve.
Then the shear strength parameters φ and c can beobtained by approximating the curved envelope to astraight line, namely the slope of the straight line equals
tan φ and the intercept of straight line on the verticalaxis is c (see Figure 2).

6

Figure 2

7

The so-called ”liquefaction phenomenon” is one of contemporary important problems especially in the country where people can not avoid large-scale earthquakes.
Therefore, Mohr failure envelope is a signiﬁcant notion
for industry.
In order to understand the mechanism of ”liquefaction
phenomenon” well and in order to ﬁnd an eﬀective measure against real liquefaction phenomena, it seems important to construct a general theory of the envelopes
created by circle families.

8326§2. Envelopes of circle families
For a point P of R2 and a positive number λ, the circle
C(P,λ) centered at P with radius λ is naturally deﬁned
as follows, where the dot in the center stands for the
standard scalar product.







C(P,λ) = (X, Y ) ∈ R2  ((X, Y ) − P ) · ((X, Y ) − P ) = λ2 .

For a curve γ : I → R2 and a positive function λ : I →R+, the circle family C(γ,λ) is naturally deﬁned as follows. Here, R+ stands for the set consisting of positive
real numbers.
C(γ,λ) = C(γ(t),λ(t))



t∈I

.

9

It is reasonable to assume that at each point γ(t) the
normal vector to the curve γ is well-deﬁned. Thus, weeasily reach the following deﬁnition.
Deﬁnition 1 A curve γ : I → R2 is called a frontal
if there exists a mapping ν : I → S 1 such that thefollowing identity holds for each t ∈ I, where S 1 is theunit circle in R2.
dγ(t) · ν(t) = 0.
dt
For a frontal γ, the mapping ν : I → S 1 given above iscalled the Gauss mapping of γ.
Hereafter, the curve γ : I → R2 for a circle family C(γ,λ)
is assumed to be a frontal.
10

In this talk, the following is adopted as the deﬁnitionof an envelope created by a circle family.
Deﬁnition 2 Let C(γ,λ) be a circle family. A mappingf : I → R2 is called an envelope created by C(γ,λ) if thefollowing two hold for any t ∈ I.
(1) dfdt (t) · (f (t) − γ(t)) = 0.
(2) f (t) ∈ C(γ(t),λ(t)).

11

327

Problem 1 Let γ : I → R2 be a frontal with Gaussmapping ν : I → S 1 and let λ : I → R+ be a positve
function.
(1) Find a necessary and suﬃcient condition for thecircle family C(γ,λ) to create an envelope in termsof γ, ν and λ.
(2) Suppose that the circle family C(γ,λ) creates an envelope. Then, ﬁnd a parametrization of the envelope created by C(γ,λ) in terms of γ, ν and λ.

12

Exampleγ : R → R2 be the mapping 1 Let deﬁned by−2t3, 1 . It is clear
γ(t) = t3, t6 . Set ν(t) = √ 16
4t +1

that the mapping γ is a frontal with Gauss mappingν : R → S 1. Let λ : R → R+ be the constant function
deﬁned by λ(t) = 1. Then, it seems that the circle
family C(γ,λ) creates envelopes. Thus, we can expectthat the created envelopes can be obtained by the wellknown method.

13

Set F (x, y, t) = x − t3
the following.

=
=
=

=

=

2



+ y − t6

2

−1 . Then, we have



∂F

(x, y) ∈ R2  ∃t s.t. F (x, y, t) =(x, y, t) = 0
∂t




2 
2





(x, y) ∈ R2  ∃t s.t. x − t3 + y − t6 − 1 = −6t2 x − t3 − 12t5 y − t6 = 0




2 
2





(x, y) ∈ R2  ∃t s.t. x − t3 + y − t6 − 1 = t2 x − t3 + 2t3 y − t6 = 0(x, y) ∈ R2  x2 + y 2 = 1


2 
2



(x, y) ∈ R2  x − t3 + y − t6 − 1 = 0, x = t3 − 2t3 y − t6
(x, y) ∈ R2  x2 + y 2 = 1



2 
2



(x, y) ∈ R2  −2t3 y − t6
+ y − t6 = 1, x = t3 1 − 2y + 2t6

(x, y) ∈ R2  x2 + y 2 = 1






2t3
1

∈ R2  t ∈ R .
t3 ∓ 
, t6 ± 
4t6 + 1
4t6 + 1


14

328

15

In order to solve Problem 1, we prepare several terminologies which can be derived from a frontal γ : I → R2
with Gauss mapping ν : I → S 1 and a positive function
λ : I → R+. For a frontal γ : I → R2 with Gauss mapping ν : I → S 1, following “T. Fukunaga and M. Takahashi, Existence and uniqueness for Legendre curves,
Journal of Geometry, 104 (2013), 297–307”, setµ(t) = J(ν(t)),
where J is the anti-clockwise rotation by π/2. Then we
have a moving frame {µ(t), ν(t)}t∈I along the frontal γ
. Set
dγ
dν
(t) · µ(t), β(t) =(t) · µ(t).
ℓ(t) =
dt
dt
16

The following deﬁnition is the key of this talk.
Deﬁnition 3 ([WN], KEY DEFINITION) Let γ : I →R2, λ : I → R+ be a frontal with Gauss mapping ν : I →S 1 and a positive function respectively. Then, the circle family C(γ,λ) is said to be creative if there existsν : I → S 1 such that the following identity holds for anyt ∈ I.
dλ(t) = −β(t) (ν(t) · µ(t)) .
dt

17

329

By deﬁnition, any family of concentric circles with smoothly
expanding radii is not creative, and it is clear that suchthe circle family does not create an envelope.
Theorem 1 ([WN]) Let γ : I → R2 be a frontal with
Gauss mapping ν : I → S 1 and let λ : I → R+ be apositive function. Then, the following hold.
(1) The circle family C(γ,λ) creates an envelope if andonly if C(γ,λ) is creative.
(2) Suppose that the circle family C(γ,λ) creates an envelope f : I → R2. Then, the created envelope f isrepresented as follows.
f (t) = γ(t) + λ(t)ν(t).

18

Example 2 We examine Example 1 by applying Theorem 1. In Example 1, γ : R → R2 is given by γ(t) =
t3, t6 . Thus, we can say that ν : R → S 1 and µ : R →S 1 are given by ν(t) = √ 16
√ 1

4t6 +1−1, −2t3



4t +1

respectively.

−2t3, 1and µ(t) =Moreover, the radiusfunction λ : R → R is the constant function deﬁned by
λ(t) = 1. Thus,dλ(t) = 0.
dt
By calculation, we haveβ(t) =dγ−3t2(1 + 4t6)
(t) · µ(t) =.
dt
4t6 + 1
19

20

330

Therefore, the unit vector ν(t) ∈ S 1 satisfyingdλ(t) = −β(t) (ν(t) · µ(t))
dtexists and it must have the form±1
ν(t) = ±ν(t) = −2t3, 1 .
4t6 + 1Hence, by the assertion (1) of Theorem 1, the circle
family C(γ,λ) creates an envelope f : R → R2.

21

By the assertion (2) of Theorem 1, f is parametrized
as follows.
f (t) = γ(t) + λ(t)ν(t)




1
−2t3, 1
= t3 , t 6 ± 
64t + 1= t 3 ∓ 

2t3



1

, t6 ± .
4t6 + 1
4t6 + 1

22

Thank you for your listening!

23

331

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Calcium waves sustained by calcium influx throughmechanically activated channels in the cell
membrane
Zbigniew Peradzyński
Military Technological University and Institute of Fundamental TechnologicalResearch PAS, Poland(joint work with Bodan Kaźmierczak and Slawomir Bialecki)
The work is devoted to the mathematical modeling of fast calcium waves propagating insome cells. According to the suggestion of biologists, this type of waves exists due to thecomplicated mechanisms of the influx of calcium from the extracellular space throughmechanically operated calcium channels placed in the cell membrane. A change in theconcentration of calcium in the cell causes the reorganization of the network composedof actin-myosin filaments. Under the influence of local forces exerted by these fibers,
ion channels in the cell membrane are opened. At the same time, excess calcium ispumped out of the cell by several types of pumps located in the cell membrane. All this
together leads to the possibility of wave propagation in the form of homoclinic pulsesof calcium concentration. We start from the construction of the model in 3-D. Then
we derive 1-D nonlocal approximation, which as it turns out, can be still approximatedby a FitzHugh Nagumo type of system. The theoretical model will also be supported
by numerical calculations.

333

Calcium waves sustained by calcium influx
through mechanically activated channels
in the cell membrane
Zbigniew Peradzyński * ,Bogdan Kaźmierczak**, Sławomir Białecki**,* Warsaw Military University of Technology (earlier inFaculty of Mathematics, Informatics and Mechanics, University of
Warsaw),**Institute of Fundamental Technological Research.
Workshop on Mathematics for Industry 2023, Warsaw

Provocative question:

Can plants be aware of the danger?
Please see the video:
https://www.youtube.com/watch?app=desktop&&v=7-3yFcZSyvo„”Supplying glutamate directly to the tip of one leaf creates a strong wave of calcium
across the entire plant, visualized by fluorescent light. This video is part of research
by UW–Madison botany professor Simon Gilroy that shows how waves of calcium
crisscrossing a plant help it respond to attacks by preparing for future threats.
The work was published in Science in September of 2018”.

It turns out that plants or their parts can communicate with each other (e.g bysending sigals calcium waves), preparing thus for unpleasant consequences334

By waves we mean travelling waves,special solutions: 𝒖𝒖 = 𝑼𝑼(𝑥𝑥 − 𝑐𝑐𝑐𝑐) toReaction-Diff. equations (c-const )• Waves are usually associated with the wave equation orwith hyperbolic systems. However hyperbolic equations
are almost nonexisting in biology. One predominantly
encounters parabolic equations or semilinear parabolic
systems – Reaction-Diffsion Systems.
• The travelling waves in R-D eqs are appearing as aninterplay between the diffusion and nonlinearity.

Single reaction –diffusion equation𝜕𝜕𝑢𝑢 = 𝐷𝐷∆𝑢𝑢 + 𝐹𝐹(𝑢𝑢)𝜕𝜕𝜕𝜕

If u(t,x) – density of individuals, F(u) = ru(1-u/K), then one canspeak of a simple model in population dynamics. The
diffusive term reflects the fact that individuals are moving
erratically. The reaction term F(u) is responsible for the birthand death processes.
Here travelling wave solutions are heteroclinic fronts. As F ismonostable, because u=0 is unstable equilibrim, there are
solutions for an arbitrary speed ≥ 𝑐𝑐0 .

Bistable case; the wave speed is uniquely determined!
F(u) has two stable: 𝑢𝑢1 , 𝑢𝑢3 and one unstable (𝑢𝑢2 ) equilibrium.
F(u)

𝑢𝑢1

𝑢𝑢2

Fig. An example of a bistable source function:
𝐹𝐹 𝑢𝑢 = −𝐴𝐴(𝑢𝑢 − 𝑢𝑢1 )(𝑢𝑢 − 𝑢𝑢2 )(𝑢𝑢 − 𝑢𝑢3 )

335

𝑢𝑢3

u

An example of a travelling front
The following bistable reaction diffusion equation witha cubic (bistable) source term𝝏𝝏𝝏𝝏𝟐𝟐𝒖𝒖 = 𝑫𝑫 𝟐𝟐 𝒖𝒖 − 𝑨𝑨 𝒖𝒖(𝒖𝒖 − 𝒂𝒂)(𝒖𝒖 − 𝟏𝟏)
𝝏𝝏𝝏𝝏
𝝏𝝏𝝏𝝏

has (D=1, A=1) following travelling front solutions𝑢𝑢 =1 + exp
12

1±𝑥𝑥 − 𝑣𝑣𝑣𝑣
2

where 𝑣𝑣 = 2 ( − 𝑎𝑎) defines the propagation speed.

Monostable reaction term – waves can propagate with an arbitrary
speed grater then some 𝑣𝑣0 . The minimal speed makes physical sense)
The case of F(u)= ru(1-u/K) is a good example of amonostable reaction term. It has two zeros:
Unstable state u=0 and stable state u=K

Just look at 𝑢𝑢𝑢 = 𝑟𝑟𝑟𝑟(1 − 𝐾𝐾/𝑢𝑢)

Theory based on single reaction diffusion equation predicts
travelling waves in the form of heteroclinic fronts, joining two
stable (in the bistable case) equilibria of the source term, whereas
observed experimentally calcium waves are of homoclinic type.
Thus, such simplified theory describes properly only the front partof the wave. To obtain the shape of a homoclinic, the additional
equation for “recovery variable” is usually added.
In the proposed here theory for CICI waves this additional equation
appears in a natural way.

336

Ecology, Population dynamics

MONOTONE SYSTEMS

Comments on a multistable case337

Calcium waves were discoveredin 1977 on medaka fish egg.
John C. Gilkey, Lionel f. Jaffe, Ellis B. Ridgway,
and George T. Reynolds „A FREE CALCIUMWAVE TRAVERSES THE ACTIVATING EGG OF
THE MEDAKA, ORYZIAS LA TIPES”,, Journ. Cell
Biology" Vol. 76, 1978

• Signals can be transmitted by various means – calcium concentration
waves among the others. After the fertilization of an egg the wave
sprading on its surface is generated, which changes the status of an
egg. The second sperm can not enter the egg.

The calcium wave through moving amoebae.
Speed 15 𝜇𝜇m/s. (L. Jaffe)338

From L. Jaffe

Calcium waves (first seen on the fertilizing medaka egg ) turned out to be quite common.
They can propagate both in individual cells and in tissues. The range of their speed:
1nm/s – 30 cm/s (nearly a billion fold) falling into four speed –based groups (after L. Jaffe)

In our lecture we will be interestet in CICI fast waves (see diagram below).

CICI WAVESThe mechanism of propagation of CICR waves is based on
autocatalytic release of calcium from the internal stores (e.g.
endoplasmic reticulum) located in the cells.

CICI waves. According to L. Jaffe this cannot explain the
speed of the second group of „fast waves” . Their speed
can be by two orders higher. Such waves are also
observed in cells not having internal stores of calcium.
Thus: Stretch-activated ion channels in the membrane areresponsible for the calcium delivery from the extracellularspace.

339

CELL is extremally complex! (Nobel Prize 2013).
The cell membrane is equipped witha) ion channels (MECHANICALLY, chemically or electrically controlled)
through which ions are admitted into the cell interior.
b) There are pumps in the membrane - at least two types:• ATP type - efficient at low 𝐶𝐶𝐶𝐶 ++ concentrations• sodium-calcium exchangers; very efficient at high 𝐶𝐶𝐶𝐶++ concentrations.
Thanks to them, balance in the cell can be restored.

Mechanically operated ion channels (stretch activated)
are opened when the membrane is stretched.

Inside the cel we have
1.
2.
3.
4.

Cytoplasm
Actin filaments
Internal stores of calcium (endoplasic reticulum)
Other important ingredients as: ion channels and ion pumps located in
the cell membrane.

• As the Ca concentration increases, the filaments are increasinglyconnected by myosin bridges and the filament network contracts.
• The filaments also serve as routes along which various materials in bags
(vesicles) are transported by appropriate motors. ( F=2.7 pN). See forexample :https://learn.genetics.utah.edu/content/cells/vesicles/

Model of a cell(from the lecture by Kizylvova)

340

Coming back to Ca waves

There are already well known and well researched CICR waves i.e.
„Calcium Induced Calcium Released” waves (L. Jaffe) . The simplest
theoretical description is based on single reaction diffusion equation
with a bistable source term. For a small excess of calcium above
the equilibrium concentration,
calcium is absorbed into internal
stores. After exceeding a certain
threshold value (the second zeroof the source function) calcium isreleased from the internal stores of the cell in an autocathalitic
reaction, untill its concentration reaches the next equilibrium value (thethird zero of source function).

341

Lionel Jaffe Hypothesis
According to L. Jaffe, the CICR mechanism cannot be responsible
for high speed of CICI waves (see diagram).
It is known that:
Stretching the membrane activates the ion channels and calcium
can enter the cell from the extracellular space.
Hypothesis: when the calcium concentration grows the actinmyosin network is reorganized – the filament network contracts.
Consequently, filaments are pulling the membrane. Mechanicallystimulated channels are opened and calcium enters the cell. Thismechanism (calcium induced calcium influx) supports the wave
propagation.

Hypothetical CICI Waves –the subject of our modelling• Accorfing to L. Jaffe in this casecalcium from the extracellular spaceenters the cell through mechanicaly
activated ion channels located in thecell membrane. In the extracellularspace 𝑪𝑪𝑪𝑪++ concentration is by twoorders higher than in the cell internal
stores. The channels are opened whenthe membrane is stretched.

Calcium pumps
Calcium pumps are ion transporters found in the cell
membrane. They are responsible for active transportof calcium out of the cell, keeping the intracellular
calcium concentration 10 000 times lower than theextracellular. The plasma membrane Ca2+ ATPaseand sodium-calcium calcium exchanger are the main
regulators of intracellular Ca2+ concentration. The
first type is efficient at low Ca concentration, whereas
the second type is extremely efficient at higher
concentrations.
They also seem to play the crucial role in supporting
the CICI Waves!

342

Assumptions.

1. The contraction of the actomyosin network results in appearing of socalled “traction forces”. However, the effect of contraction following theincrease of calcium concentration appears with some delay –relaxation timeis needed to form the myosin bridges
2. The calcium can enter from the intercellular space through themechanically stimulated ion channels located in the cell membrane
3. The mechanical stimulation of the membrane is caused by theactomyosin network - cortex. The fibers of the cortex as well as the rest ofactomyosin network in the cell are subject to the contraction whenever the
calcium concentration in the cell cytoplasm increases.

As the calcium concentration increases, the myosin
filaments become more and more connected through theincreasing number of myosin Bridges. This leads to thecontraction of the filament network.
This contraction influences the shape of the cell. If we
imagine the ideal cell of a cylindrical shape, then the cellradius will be reduced. Therefore, at first glance, we should
not expect any stretching of the cell membrane.
This is however macroscopic view. Microscopically themembrane will be very unsmooth. Funnel-shaped
depressions will appear under the influence of pulling
forces, in places where the filaments are anchored. So in
spite of this that the average radiuce gets smaller we will
have the membrane stretching as its shape becames more
complex.

When the wave passes, the cell radius shrinks. So how can
we have stretching ? locally we expect the following picture343

Suppose, the ion channels are openned whenever themembrane is streched. Then permanent stretch :High calcium concentration over a long period of time would
lead to the cell death. Therefore, a permanent state of stretch
should not result in a continuous influx of calcium.
Experiment: oscillatory stretching leads to 𝐶𝐶𝐶𝐶++ influx
proportional to the amplitude and oscillations frequency.

This suggests that the calcium influx should rather be related tothe speed of membrane stretching !

H.1. Therefore, if n is an internal unit vector normal to the cell
membrane and F is the force acting on unit membrane area,
then the calcium influx (flux per unit area) is proportional tothe positive part of the time derivative of the force acting on
the unit surface.
𝐶𝐶𝐶𝐶++ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ~𝜕𝜕(𝒏𝒏 ∙ 𝑭𝑭)𝜕𝜕𝜕𝜕+

Positive part, because only stretching counts. One can showthat otherwise the Ca concentration may become negative !

Taking into account the pumps p(c)𝑝𝑝 𝑐𝑐 :

𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐶𝐶𝐶𝐶++ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ~𝜕𝜕(𝒏𝒏 ∙ 𝑭𝑭) − 𝑝𝑝(𝑢𝑢) 𝜕𝜕𝜕𝜕+This is the boundary condition for the Ca diffusion equation.

344

Now we arrived at the MECHANICAL PROBLEM:
Determine the forces acting on the membrane ; i.e. forces
resulting from the actin filaments attached to it.
In principle two approaches seem to be possible:a) Calculate the distribution of forces on each filament of thecontracting network due to the appearance of myosin
bridges. In particular those anchored in the membrane. Thenfind the shape of deformed membrane.

This seems hopelessly difficult !

Continuum mechanical approach ?
b) In mathematical biology (Murray, Mathematical Biology) ,the cell is often treated as an elastic (or viscoelastic) body, andthe forces associated with the contraction (traction forces) are
expressed by the traction tensor. This description is very
similar to termo-elasticity. 𝑪𝑪𝑪𝑪++ concentration plays the roleof the temperature (in fact −𝑻𝑻).
Applying this idea, we arrive at a system of three equations.

The system consists of
1. The equation of motion of the viscoelastic body, i.e cytoplasm withthe filament network. The equation of motion (linear approximation) for
the displacement vector 𝒖𝒖(𝒕𝒕, 𝒙𝒙) must be equipped with properboundary conditions. Under the influence of traction forces themembrane is deflected. So basically, we should know the elasticity ofthe membrane. However, to estimate the forces acting on themembrane, one can assume that the membrane is stiff and notdeformed. In such a case we have simple b-dry condition: u(R) = 0

Let us remind that if the initial position of a material point is 𝒙𝒙 and it෥ − 𝒙𝒙.
෥ then 𝒖𝒖 𝒙𝒙 = 𝒙𝒙
position changes to 𝒙𝒙345෡ with a given equilibrium
2. Relaxation equation for the traction tensor 𝑻𝑻෡ ∗ 𝒄𝒄 . We have 𝑻𝑻෡ (𝑡𝑡, 𝒙𝒙) = 𝑻𝑻෡ ∗ 𝒄𝒄(𝑡𝑡, 𝒙𝒙) for very slow changes of theform 𝑻𝑻
concentration 𝒄𝒄(𝒕𝒕, 𝒙𝒙).

3. The diffusion equation for calcium concentration 𝒄𝒄(𝑡𝑡, 𝒙𝒙) and nonlinear
boundary condition expressing the influx of calcium (by ion channels and ion
pumps) caused by positive part of time derivative of traction forces acting on
the membrane.
In fact, the diffusion of calcium in the cell is quite a complicated process
because of the buffers - proteins that can attach and release calcium ions. Thiscan be described by a system of equations for the diffusion reaction. If we use
one equation as here, D should be treated as the effective diffusion
coefficient.

Treating (idealized) cell as an Infinite cylinder we could try to solve:(1) 𝝆𝝆

𝝏𝝏𝟐𝟐𝒖𝒖 − 𝝂𝝂𝟐𝟐 ∆ 𝒖𝒖ሶ +𝝏𝝏𝒕𝒕𝟐𝟐෡ (𝒄𝒄)
𝝂𝝂𝟏𝟏 + 𝝂𝝂𝟐𝟐 𝛁𝛁𝛁𝛁𝛁𝛁𝛁𝛁𝒖𝒖ሶ =μ ∆ 𝒖𝒖 + 𝝁𝝁 + 𝝀𝝀 𝜵𝜵𝜵𝜵𝜵𝜵𝜵𝜵𝜵𝜵 + 𝒅𝒅𝒅𝒅𝒅𝒅 𝑻𝑻with b-dry condition: 𝒖𝒖 𝑡𝑡, 𝑅𝑅 = 0𝜕𝜕෡ = 𝛽𝛽[𝑻𝑻෡ ∗ 𝒄𝒄
𝑻𝑻
𝜕𝜕𝜕𝜕

(2)

(3)

𝐷𝐷

𝜕𝜕
𝑐𝑐 = 𝐷𝐷∆𝑐𝑐𝜕𝜕𝜕𝜕𝜕𝜕
𝑐𝑐 𝑡𝑡, 𝑅𝑅, 𝑧𝑧
𝜕𝜕𝜕𝜕

෡ ],
− 𝑻𝑻

෡ ∗ 𝒄𝒄 - known (e.g. linear)where 𝑻𝑻

inside the cell

= 𝑄𝑄

+
𝑑𝑑
𝜎𝜎 (𝑡𝑡, 𝑅𝑅, 𝑧𝑧) − 𝑝𝑝(𝑢𝑢)𝑑𝑑𝑑𝑑 𝑟𝑟𝑟𝑟

on the b-dry

suplied by initial conditions for 𝑢𝑢, 𝑇𝑇, 𝑐𝑐 .

Comment. The first equation, the equation of motion can be𝝏𝝏𝟐𝟐

simplified by omitting the dynamical term 𝝆𝝆 𝟐𝟐 𝒖𝒖 and possibly the𝝏𝝏𝒕𝒕viscouse terms 𝝂𝝂𝟐𝟐 ∆ 𝒖𝒖ሶ + 𝝂𝝂𝟏𝟏 + 𝝂𝝂𝟐𝟐 𝛁𝛁𝛁𝛁𝛁𝛁𝛁𝛁𝒖𝒖ሶ .

Then one obtains an eliptic system for the displacement 𝒖𝒖(𝑡𝑡, 𝑥𝑥).

In principle it is possible to solve the above system numerically. For
reasons discussed below, we decided on a slightly roundabout but
simpler route.

346

In presented here equations we assumed the medium to be isotropic.
However, the anisotropy, can be important as it can greatly influence thespeed of waves. Indeed, the network structure - the way the filaments areconnected, affects the transfer of forces acting on the membrane throughthe interconnected fibers.
Depending on the way the filament network is interconnected, calcium
channels may be opened in places more or less distant from the front of the
wave of increased calcium concentration. Thus, we should solve systems
with different degree of anisotropy.
To avoid all these complications, we chose a slightly different modelingroute.

෡ = 𝜏𝜏I
Intermediate way, Here 𝑻𝑻

Instead, we chose the intermediate solution. By solving the equationsof mechanical equilibrium,෡ 𝒄𝒄 = 𝟎𝟎
μ ∆ 𝒖𝒖 + 𝝁𝝁 + 𝝀𝝀 𝜵𝜵𝜵𝜵𝜵𝜵𝜵𝜵𝜵𝜵 + 𝒅𝒅𝒅𝒅𝒅𝒅 𝑻𝑻

assuming that the solution is independent of the axial variable, and for෡ = 𝜏𝜏𝐈𝐈 we can estimate the forces acting on
isotropic traction tensor 𝑻𝑻
the membrane as
𝑅𝑅

1
𝜎𝜎𝑟𝑟𝑟𝑟 𝑡𝑡, 𝑅𝑅 =න 𝜏𝜏 𝑡𝑡, 𝑟𝑟 2𝜋𝜋𝑟𝑟𝑑𝑑𝑟𝑟
𝜋𝜋𝜋𝜋20Since the Ca influx is proportional to time derivative of 𝜎𝜎𝑟𝑟𝑟𝑟
𝜕𝜕
𝜎𝜎 𝑡𝑡, 𝑅𝑅
𝜕𝜕𝑡𝑡 𝑟𝑟𝑟𝑟

we have=𝑅𝑅 𝜕𝜕1𝜏𝜏 𝑡𝑡, 𝑟𝑟 2𝜋𝜋𝑟𝑟𝑑𝑑𝑟𝑟‫׬‬𝜋𝜋𝜋𝜋 2 0 𝜕𝜕𝜕𝜕𝜕𝜕𝜏𝜏 = 𝛽𝛽 [𝜏𝜏 ∗ 𝑐𝑐𝜕𝜕𝜕𝜕𝑅𝑅− 𝜏𝜏] , so𝜕𝜕𝛽𝛽
න [𝜏𝜏 ∗ 𝑐𝑐 − 𝜏𝜏] 2𝜋𝜋𝑟𝑟𝑑𝑑𝑟𝑟
𝜎𝜎𝑟𝑟𝑟𝑟 𝑡𝑡, 𝑅𝑅 =
𝜕𝜕𝑡𝑡
𝜋𝜋𝜋𝜋2
0

347

Smearing the force (interconnected filaments)
The previous step do not include transmition of force from one point toanother by interconnected filaments. To take this into account we introduce akind of smearing out of forces acting on the membrane through an averaging
integral operator (convolution wit 𝐾𝐾𝜎𝜎 )𝐷𝐷𝜕𝜕𝑐𝑐 𝑡𝑡, 𝑅𝑅. 𝑧𝑧 = 𝐴𝐴𝜕𝜕𝜕𝜕

𝐾𝐾𝜎𝜎 ∗

𝑅𝑅

2
න 𝜏𝜏 ∗ (𝑐𝑐 𝑡𝑡, 𝑟𝑟, 𝑧𝑧 ) − 𝜏𝜏 𝑡𝑡, 𝑟𝑟, 𝑧𝑧
𝑅𝑅20

where in numerical simmulations we took 𝐾𝐾𝜎𝜎 =𝑟𝑟𝑟𝑟𝑟𝑟+− 𝑝𝑝(𝑐𝑐)1𝑧𝑧 2
exp(− 2 ).
𝜎𝜎 2𝜋𝜋2𝜎𝜎This non-local mechanism embodies the idea of ​L. Jaffe

membraneactin network

Schematic view of simplest model of actin fibers network in 2D.
When Ca concentration increases the fibers contract pullingthe membrane. This arrangment of fibers corresponds to completely
anisotropic case ( no myosin bridges between filaments). The force is nottransfered between filaments – local mechanism.
𝐾𝐾~ 𝛿𝛿(𝑥𝑥)membraneactin network

cortexNonlocalmechanism of
propagation• This mechanizm is nonlocal. The filaments
are interconnected by myosin bridges.
Their number grows with Ca concentration.
• The force that appears in one place istransferred by the tangled fibers to otherneighboring ones. Thus the channels areopenned ahead of the propagating wave of
𝐶𝐶𝐶𝐶++ concentration. This accelerates thewave propagation.

348

Numerical computations
All numerical computations were done for the diffusion coefficient D=1.
The source term:[𝑲𝑲 𝟎𝟎, 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟎𝟎. 𝟏𝟏𝒖𝒖𝟐𝟐 − 𝝉𝝉 ]+ −𝒑𝒑(𝒖𝒖) where

𝒑𝒑 𝒖𝒖 = 𝒖𝒖 𝒖𝒖𝟐𝟐 − 𝟏𝟏. 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟎𝟎. 𝟓𝟓
For K=id and 𝜏𝜏 ≡ 0 the source term takes form
𝑢𝑢 𝑢𝑢 − 0,25)(𝑢𝑢 − 1
𝜕𝜕

𝜕𝜕2

Eq.
𝑢𝑢 = 2 𝑢𝑢 − 𝑢𝑢 𝑢𝑢 − 0,25)(𝑢𝑢 − 1 has heteroclinic solutions
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
(travelling fronts) of the form

Source term for 𝜏𝜏 = 0
For 𝜏𝜏 = 0 we must have bistable case!

3D NUMERICAL SIMMULATIONS l
Assuming cylindical symmetry we solved numerically the system :𝝏𝝏𝒄𝒄 = 𝑫𝑫∆𝒄𝒄𝝏𝝏𝝏𝝏

in Ω ,

𝛛𝛛𝛕𝛕 = 𝛃𝛃[𝛕𝛕∗ 𝐜𝐜
𝛛𝛛𝛛𝛛

− 𝛕𝛕]

𝑫𝑫 𝒏𝒏 ∙ 𝛁𝛁𝛁𝛁 = 𝑨𝑨{ 𝑲𝑲𝝈𝝈𝛛𝛛𝛕𝛕 − 𝒑𝒑 𝒄𝒄 }𝛛𝛛𝛛𝛛 +349

in Ω

on 𝜕𝜕Ω,Numerically determined travelling homoclinic waves (moving to the right)
𝐶𝐶𝐶𝐶++ concentration (for different 𝜎𝜎)

Numerical computations
All numerical computations were done for the diffusion coefficient D=1.
The source term:[𝑲𝑲 𝟎𝟎, 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟎𝟎. 𝟏𝟏𝒖𝒖𝟐𝟐 − 𝝉𝝉 ]+ −𝒑𝒑(𝒖𝒖) where

𝒑𝒑 𝒖𝒖 = 𝒖𝒖 𝒖𝒖𝟐𝟐 − 𝟏𝟏. 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟎𝟎. 𝟓𝟓
For K=id and 𝜏𝜏 ≡ 0 the source term takes form
𝑢𝑢 𝑢𝑢 − 0,25)(𝑢𝑢 − 1
𝜕𝜕

𝜕𝜕2

Eq.
𝑢𝑢 = 2 𝑢𝑢 − 𝑢𝑢 𝑢𝑢 − 0,25)(𝑢𝑢 − 1 has heteroclinic solutions
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
(travelling fronts) of the form

ONE DIMENSIONAL APPROXIMATION

Averaging our diffusion equation with respect to r :and similarly, the equation for the traction, we arrive at
the one dimensional problem
𝝏𝝏
𝝏𝝏𝟐𝟐
𝟐𝟐𝟐𝟐
𝒖𝒖 = 𝑫𝑫 𝟐𝟐 𝒖𝒖 +𝛃𝛃𝐊𝐊 𝟐𝟐 ∗ [𝛕𝛕∗ 𝐮𝐮
𝝏𝝏𝝏𝝏
𝑹𝑹
𝝏𝝏𝝏𝝏

where

𝛛𝛛
𝛕𝛕 = 𝛃𝛃[𝛕𝛕∗ 𝐮𝐮
𝛛𝛛𝛛𝛛𝑢𝑢(𝑡𝑡, 𝑥𝑥) =− 𝛕𝛕]− 𝛕𝛕] − 𝒑𝒑(𝒖𝒖)𝑅𝑅1‫ ׬‬2𝜋𝜋𝜋𝜋 𝑐𝑐 𝑡𝑡, 𝑥𝑥, 𝑟𝑟 𝑑𝑑𝑑𝑑
𝜋𝜋𝑅𝑅 2 0350

Waves profiles at r=R, (R=2) for different β: (a) β=0,1𝜷𝜷𝟎𝟎 , (b) β=0,2𝜷𝜷𝟎𝟎 , (c) β=0,3𝜷𝜷𝟎𝟎
etc. where the reference 𝜷𝜷 is 𝜷𝜷𝟎𝟎 = 𝟎𝟎, 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎. On the left for σ=10. On the rightfor σ=20.

On the left: wave profiles and wave velocities in 1-D simulations for A=1, and (a) 𝜎𝜎=40, (b) 𝜎𝜎=20, (c)𝜎𝜎=10, (d) 𝜎𝜎=0 and for 𝛽𝛽=0.001205 (=0.1 𝛽𝛽0 ). On the right: 3D simulations for A=1, R=2, and β=0,001205
and the same values of σ. Propagation velocities with respect to the heteroclinic case ( 𝑣𝑣0 = 2/4) are:(a)13.7,(b) 6.96, (c) 3.63 , (d) 0.978

Fitzhugh –Nagumo type of approximation
The influence of the variance 𝜎𝜎 of 𝐾𝐾𝜎𝜎 on the wave velocity.
Expanding :𝐊𝐊 𝟐𝟐 ∗ 𝛕𝛕∗ 𝐮𝐮 we arrive to easier, local system of equations
𝝏𝝏
𝝏𝝏𝟐𝟐
𝑨𝑨
𝒘𝒘 = 𝟐𝟐 𝑫𝑫𝑫𝑫 + 𝝈𝝈𝟐𝟐 𝝉𝝉∗ 𝒘𝒘
𝝏𝝏𝝏𝝏
𝝏𝝏𝝏𝝏
𝑹𝑹
𝝏𝝏
𝝉𝝉 = 𝜷𝜷[𝝉𝝉∗ 𝒘𝒘 − 𝝉𝝉]
𝝏𝝏𝝏𝝏

+

𝟐𝟐𝟐𝟐
𝑹𝑹

𝝉𝝉∗ 𝒘𝒘 − 𝝉𝝉 + − 𝒑𝒑(𝒘𝒘)with larger diffusivity. The wave velocity is ~ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑351

F-N model is simple and gives good wave speed.
This F-N model we studied (with J. Napiokowska) for a particular shapeof the source term step like 𝝉𝝉∗ 𝒘𝒘 and linear 𝒑𝒑(𝒘𝒘).
• In this case the existence of homoclinic waves is proven for somerange of 𝛽𝛽 < 𝛽𝛽0 ,• For 𝛽𝛽 > 𝛽𝛽0 there are no homoclinic waves.
• There are two solutions for given 𝛽𝛽 < 𝛽𝛽0 . Narrow one unstable andwider which is stable.

Conclusions
1. It seems that the idea of F. Jaffe worksa) Wave velocity grows as 𝜎𝜎 . 𝜎𝜎 – range of mechanicalinteractions due to actin-myosin fiber network.
b) The concentration of Ca in extracellular space is
100 times bigger than in endoplasmic reticulum, so
flux through ion channel can be quite high. Again,
wave velocity grows as 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆
2. 1-D approximation seems to work quite well ! It
well reproduces the 3-D simulations.

REFERENCES[1] L.F. Jaffe, "Stretch-activated calcium channels relay fast calcium waves
propagated by calcium-induced calcium inux", Biol. Cell 99, 175{184 (2007).
[2] J.C. Gilkey, L.F. Jaffe, E.B. Ridgway, G.T. Reynolds GT, "A free calcium wave
traverses the activating egg of the Medaka, Oryzias latipes", J. Cell Biol. 76, 448466 (1978).
[3]Z. Peradzynski, B. Kaźmierczak, S. Białecki, Modelling fast CICI Waves – to be
published.
[4] Murray JD. Mathematical Biology, Springer, Berlin, 2nd edition, 1993.
[5] W.S. Nishitani, Molecular mechanisms involved in cell response to mechanicalforces, phd dissertation, University of Illinois at Urbana-Champaign, 2011.
[6] K. Okeyo, et al., "Actomyosin contractility spatiotemporally regulates actin
network dynamics in migrating cells", J. Biomech. 42(15), 2540-8 (2009).
[7] Z. Peradzynski, "Diffusion of calcium in biological tissues and accompanying
mechanochemical effects", Arch. Mech. 62, 423-440 (2010).
[8] B. Kaźmierczak, Z. Peradzyński, Calcium waves with fast buffers and mechanicaleffects, J. Math. Biol., DOI 10.1007/s00285-009-0323-2, 2010.

352

Thank you for your attention
andthe organizers for the invitation.

353

WORKSHOP on Mathematics for Industry 2023
September 25-29, 2023, Warsaw, Poland

Generalization of Reeb spaces and application todata visualization
Osamu Saeki
Institute of Mathematics for Industry, Kyushu University, Japan
In many cases, data sets can be considered to be discrete samples of differentiable maps
between manifolds. For a differentiable multivariate function into Rp with p ≥ 2, its
Reeb space is the space of connected components of its fibers. This is a generalizationof the notion of Reeb graphs for univariate functions in the case of p = 1. It has been
known that Reeb spaces are often very useful for visualizing the given multivariate
function. In this talk, we generalize the Reeb space in such a way that it captures
more of the topological features of the fibers, not only their connected components.
This theoretical part essentially relies on the global singularity theory of differentiablemaps between manifolds developed mainly by the author. Such techniques have beenused for efficiently visualize large scale data. If time permits, we will also discuss anapplication to multi-objective optimization problems.

355

Generalization of Reeb Spaces andApplication to Data Visualization
Osamu Saeki(Institute of Mathematics for Industry,
Kyushu Univ., Japan)

September 29, 2023
WORKSHOP on
Mathematics for Industry
Warsaw University of Technology

Who am I?
§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Got PhD in Mathematics in 1992.
“On 4-manifolds homotopy equivalent to the 2-sphere”

Saeki

1
LNM
1854

Lecture Notes in Mathematics

Main interest: Singularity Theory, 3- and 4-Dimensional Topology
I proposed the Theory of Singular Fibers of Diﬀerentiable Maps.
κ=1

Osamu Saeki

κ=2Topology of Singular Fibersof Differential Maps

1854

Topology of Singular Fibers of Differential Maps

κ=3

1 23

My recent interests include collaboration with computer scientists onenhancing visualization of multi-variate data from the viewpoint of
topology or singularity theory.

2 / 23§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond§1. Fiber

356

Setting§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

N n : C ∞ manifold (e.g. bdd domain in Rn ), f : N n → Rm C ∞ map
We can write f = (f1 , f2 , . . . , fm ) multi-variate data
We assume f is generic (C ∞ stable, C 0 stable, ﬁnite codimension, etc.)
We are interested in the topology of ﬁbers f −1 (y), y ∈ Rm .
Generically, dim f −1 (y) = n − m. We usually assume n ≥ m.
NRf
We can grasp global feature of data by chasing ﬁbers (or level sets).
We have singular ﬁbers (or critical level sets) where topologicaltransitions of ﬁbers occur.
4 / 23Singular points and Jacobi set§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

f : N n → Rm (n ≥ m) C ∞ map
For p ∈ N n , consider the diﬀerential dfp : Tp N n → Tf (p) Rm .
The set of singular points J(f ) = {p ∈ N n | rank dfp < m} is called theJacobi set of f . Generically, dim J(f ) = m − 1.
Jacobi set image f (J(f )) divides the range Rm into some regions.

Topology of ﬁbers changes along the Jacobi set image.
Singular ﬁber is a ﬁber f −1 (y) with y ∈ f (J(f )).
It is important to know topological changes of ﬁbers near a singular ﬁber.
5 / 23

Classiﬁcation of singular ﬁbers§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

For certain dimensions, we can classify singular ﬁbers of generic maps.
Example 1.1 Classification results for (n, m) with n − m = 1.
For simplicity, we assume the domain N n is orientable.
We will ignore regular fiber components.
1. (n, m) = (2, 1) [Folklore]κ = 1 (codimension)

2. (n, m) = (3, 2) [Kushner–Levine–Porto, 1984]
κ=1

κ=2

6 / 23357

Singular ﬁbers for (n, m) = (4, 3)§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

3. (n, m) = (4, 3) [S.]
κ=1
κ=2

κ=3

4. (n, m) = (5, 4) [Yamamoto–S.]
7 / 23§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond§2. Reeb Space

Reeb space§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

For a C ∞ map f : N n → Rm , n > m, the space Rf obtained by contracting
each connected component of a ﬁber to a point is called the Reeb space off [Edelsbrunner–Harer–Patel, 2008].
Nn
Rm
f

f¯

qf

Rf
When m = 1, it is also called the Reeb graph.
9 / 23358

Local structures of Reeb spaces§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Classiﬁcation of ﬁbers⇒ Characterization of local structures of Reeb spaces
Example 2.1 1. (n, m) = (2, 1)

2. (n, m) = (3, 2) [Kushner-Levine-Porto, 1984]

3. (n, m) = (4, 3) [Hiratuka, 2001]
10 / 23

Applications§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

An example of an application of Reeb graphs:[Takahashi–Takeshima–Fujishiro, 2004] Topological Volume
Skeletonization and its Application to Transfer Function Design
See Fig. 4 of [STSWKCDY].
Explicit example: Atom collision
See Fig. 5 of [STSWKCDY].
Application of singular ﬁbers: Hurricane Isabel data
See Fig. 15 of [STSWKCDY].
Reference:[STSWKCDY] O. Saeki, S. Takahashi, D. Sakurai, Hsiang-Yun Wu,
K. Kikuchi, H. Carr, D. Duke, and T. Yamamoto, Visualizing multivariate
data using singularity theory, The Impact of Applications on Mathematics,
Proceedings of Forum “Math-for-Industry” 2013, pp.51–65, Springer, 2014.
11 / 23

User interface§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Implemented by Daisuke Sakurai (Kyushu Univ.)
12 / 23359

§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond§3. Case with n − m = 2 andBeyondRegular ﬁbers for n − m = 2§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

How about the case n − m = 2 ?
We assume the domain N n is orientable, compact, and w/o boundary.
Regular ﬁbers are closed orientable surfaces.
ΣggΣg : closed orientable surface of genus g

14 / 23

Reeb graph for (n, m) = (3, 1)§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Case with (n, m) = (3, 1).
Σg
Σg+1
S 2 = Σ0g
Σg1 +g2
Σg1

Σg2

g1

g2
15 / 23360

Reeb space for (n, m) = (4, 2)§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Each regular “stratum” has its own label (genus of the corresponding regularﬁber component). [Furuya, 1986]
g2
g3
g1
g1 = g2 + g3

0
g+2
g+1

g+1

g1

g

g3
g2

g+1

g1

g3

g2

g2
g1

g1

g1 + 1
g1 = g2 + g3

g

g2 + 1

g1 + 1
g1 = g2 + g3

0

g

g

g3
g4

g1 + g4
g2 = g3 + g4
g = g1 + g2

g+1

g

16 / 23

0-th Homology§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Reeb space describes the connected components of ﬁbers and their“adjacencies”.

y1 y y2
I
H0 (f

−1

⏐
⏐
∼
=�

(y1 )) −−−→ H0 (f −1 (I)) ←−−− H0 (f −1 (y2 ))
⏐
⏐
∼⏐
∼⏐
=
=
�
�
H0 (f −1 (y))
⏐
⏐
∼
=�

Z⊕Z

Z

Z

17 / 23

Reeb diagram§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

This idea can be extended to homology groups of any dimension (orhomotopy groups, if you want).
Hi (f −1 (y1 )) → Hi (f −1 (y)) ← Hi (f −1 (y2 ))aαHi (a)Hi (α)βb

Hi (b)γHi (β)cdδHi (c)

Hi (δ)Hi (d)Hi (γ)=⇒ Notion of Reeb diagram.
18 / 23361

Categorical formulation§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

K1
H2
K2

H1
K4

G
H3

H4
K3
H1
K1

G

K2

H2
In a certain categorical formulation of a Reeb space, this can be considered
to be a functor.
Remark 3.1 This makes sense if each stratum is contractible.
19 / 23

Monodromy§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

If a stratum is not contractible, we need to consider monodromy.
Suppose π : E → B is a C ∞ ﬁber bundle with ﬁber S.
MCG(S) = π0 (Diﬀ + (S)) mapping class group
Associated to π is the monodromy π1 (B, b0 ) → MCG(S).
This measures the “twist” of the ﬁbers along a loop in the base B.

S

E
π
B

b0

20 / 23

Monodromy in Reeb diagram§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Given a generic map f : N n → Rm , we can subdivide Rm (or the Reeb space
Rf ) so that each stratum is contractible.
In this case, the monodromy is hidden in the Reeb diagram.

K1

H1

K2

H2
K3

H6

H3
K4

K6
H5

K5

H4

π1 (B, b0 ) → MCG(S) → Aut(H∗ (S))

Problem 3.2 Formulate all these, including monodromy! Category theory?
How to compute Reeb diagram and/or monodromy?
21 / 23

362

Possible application§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Let us consider a multi-ojective optimization problem.
Such a problem can be formulated in terms of a C ∞ muti-functionf = (f1 , f2 , . . . , fm ) : N n → Rm .

For example, given a bench-mark problem of multi-optimization, we canevaluate its complexity or certain characteristics in terms of its Reeb space,or more generally, its Reeb diagram.

22 / 23

Ending§1. Fiber §2. Reeb Space §3. Case with n − m = 2 and Beyond

Thank you for your attention!
Acknowledgements to collaborators:
Takahiro Yamamoto (Tokyo Gakugei Univ.),
Daisuke Sakurai (Kyushu Univ.),
Shigeo Takahashi (Univ. of Aizu),
Hamish Carr & David Duke (Univ. of Leeds),
Hsiang-Yun Wu (St. Pölten Univ. of Applied Sciences),
Keisuke Kikuchi

23 / 23363

MI レクチャーノートシリーズ刊行にあたり
本レクチャーノートシリーズは、文部科学省 21 世紀 COE プログラム「機能数理学の構築と展開」（H15-19 年度）において作成した COE Lecture Notesの続刊であり、文部科学省大学院教育改革支援プログラム「産業界が求める数学博士と新修士養成」（H19-21 年度）および、同グローバル COE プログラ
ム「マス・フォア・インダストリ教育研究拠点」（H20-24 年度）において行われた講義の講義録として出版されてきた。平成 23 年 4 月のマス・フォア・イ
ンダストリ研究所（IMI）設立と平成 25 年 4 月の IMI の文部科学省共同利用・共同研究拠点として「産業数学の先進的・基礎的共同研究拠点」の認定を受け、今後、レクチャーノートは、マス・フォア・インダストリに関わる国内外の研究者による講義の講義録、会議録等として出版し、マス・フォア・インダ
ストリの本格的な展開に資するものとする。
2022 年 10 月
マス・フォア・インダストリ研究所
所長

梶原 健司

International Project Research-Workshop (I)WORKSHOP on Mathematics for Industry

Basis of Mathematics in nanomedicine structures and life sensing
発
編

行
集

発

行

印

刷

2024年 3 月 14 日
Osamu Saeki, Wojciech Domitrz, Stanisław Janeczko, Marcin Zubilewicz,
Michał Zwierzyński
九州大学マス・フォア・インダストリ研究所
九州大学大学院数理学府〒819-0395 福岡市西区元岡744
九州大学数理・IMI 事務室
TEL 092-802-4402 FAX 092-802-4405
URL https://www.imi.kyushu-u.ac.jp/
城島印刷株式会社〒810-0012 福岡市中央区白金 2 丁目 9 番 6 号
TEL 092-531-7102 FAX 092-524-4411

シリーズ既刊
Issue

Author／Editor

Title

Published

COE Lecture Note

Mitsuhiro T. NAKAO
Kazuhiro YOKOYAMA

Computer Assisted Proofs - Numeric and Symbolic Approaches 199pages

August 22, 2006

COE Lecture Note

M.J.Shai HARAN

Arithmetical Investigations - Representation theory, Orthogonal
polynomials and Quantum interpolations- 174pages

August 22, 2006

COE Lecture Note Vol.3

Michal BENES
Masato KIMURA
Tatsuyuki NAKAKI

Proceedings of Czech-Japanese Seminar in Applied Mathematics 2005
155pages

COE Lecture Note Vol.4

宮田

辺要素有限要素法による磁界解析 - 機能数理学特別講義 21pages May 15, 2007

COE Lecture Note Vol.5

Francois APERY

Univariate Elimination Subresultants - Bezout formula, Laurent series
and vanishing conditions - 89pages

September 25, 2007

COE Lecture Note Vol.6

Michal BENES
Masato KIMURA
Tatsuyuki NAKAKI

Proceedings of Czech-Japanese Seminar in Applied Mathematics 2006
209pages

October 12, 2007

COE Lecture Note Vol.7

若山 正人
中尾 充宏

九州大学産業技術数理研究センター キックオフミーティング
138pages

October 15, 2007

COE Lecture Note Vol.8

Alberto PARMEGGIANIIntroduction to the Spectral Theory of Non-Commutative Harmonic
Oscillators 233pages

January 31, 2008

COE Lecture Note Vol.9

Michael I.TRIBELSKY

Introduction to Mathematical modeling 23pages

February 15, 2008

COE Lecture Note Vol.10

Jacques FARAUT

Infinite Dimensional Spherical Analysis 74pages

March 14, 2008

COE Lecture Note Vol.11

Gerrit van DIJK

Gelfand Pairs And Beyond 60pages

August 25, 2008

COE Lecture Note Vol.12

Faculty of Mathematics,
Kyushu University

Consortium “MATH for INDUSTRY” First Forum 87pages

September 16, 2008

COE Lecture Note Vol.13

九州大学大学院
数理学研究院

プロシーディング「損保数理に現れる確率モデル」― 日新火災・九州大学 共同研究2008年11月 研究会 ― 82pages

February 6, 2009

健治

October 13, 2006

シリーズ既刊
Issue

Author／Editor

Title

Published

COE Lecture Note Vol.14

Michal Beneš,
Tohru Tsujikawa
Shigetoshi Yazaki

Proceedings of Czech-Japanese Seminar in Applied Mathematics 2008
77pages

February 12, 2009

COE Lecture Note Vol.15

Faculty of Mathematics,
Kyushu University

International Workshop on Verified Computations and Related Topics
129pages

February 23, 2009

COE Lecture Note Vol.16

Alexander Samokhin

Volume Integral Equation Method in Problems of Mathematical Physics February 24, 2009
50pages

COE Lecture Note Vol.17

矢嶋
及川
梶原
辻
福本

非線形波動の数理と物理 66pages

February 27, 2009

COE Lecture Note Vol.18

Tim Hoffmann

Discrete Differential Geometry of Curves and Surfaces 75pages

April 21, 2009

COE Lecture Note Vol.19

Ichiro Suzuki

The Pattern Formation Problem for Autonomous Mobile Robots―Special Lecture in Functional Mathematics― 23pages

April 30, 2009

COE Lecture Note Vol.20

Yasuhide Fukumoto
Yasunori Maekawa

Math-for-Industry Tutorial: Spectral theories of non-Hermitian
operators and their application 184pages

June 19, 2009

COE Lecture Note Vol.21

Faculty of Mathematics,
Kyushu University

Forum “Math-for-Industry”
Casimir Force, Casimir Operators and the Riemann Hypothesis
95pagesNovember 9, 2009

COE Lecture Note Vol.22

Masakazu Suzuki
Hoon Hong
Hirokazu Anai
Chee Yap
Yousuke Sato
Hiroshi Yoshida

The Joint Conference of ASCM 2009 and MACIS 2009:
Asian Symposium on Computer Mathematics Mathematical Aspects ofComputer and Information Sciences 436pages

December 14, 2009

COE Lecture Note Vol.23

荒川 恒男
金子 昌信

多重ゼータ値入門

February 15, 2010

COE Lecture Note Vol.24

Fulton B.Gonzalez

Notes on Integral Geometry and Harmonic Analysis

COE Lecture Note Vol.25

Wayne Rossman

Discrete Constant Mean Curvature Surfaces via Conserved Quantities
130pages

May 31, 2010

COE Lecture Note Vol.26

Mihai Ciucu

Perfect Matchings and Applications

July 2, 2010

徹
正行
健司
英一
康秀

111pages

66pages

125pages

March 12, 2010

シリーズ既刊
Issue

Author／Editor

Title

Published

COE Lecture Note Vol.27

九州大学大学院
数理学研究院

Forum “Math-for-Industry” and Study Group Workshop
Information security, visualization, and inverse problems, on the basisof optimization techniques 100pages

October 21, 2010

COE Lecture Note Vol.28

ANDREAS LANGERMODULAR FORMS, ELLIPTIC AND MODULAR CURVESLECTURES AT KYUSHU UNIVERSITY 2010 62pages

November 26, 2010

COE Lecture Note Vol.29

木田 雅成
原田 昌晃
横山 俊一

Magma で広がる数学の世界

December 27, 2010

COE Lecture Note Vol.30

原
松井
廣島

Mathematical Quantum Field Theory and Renormalization Theory
201pages

January 31, 2011

COE Lecture Note Vol.31

若山 正人
福本 康秀
高木
剛
山本 昌宏

Study Group Workshop 2010 Lecture & ReportFebruary 8, 2011

COE Lecture Note Vol.32

Institute of Mathematics
for Industry,
Kyushu University

Forum “Math-for-Industry” 2011
“TSUNAMI-Mathematical Modelling”
Using Mathematics for Natural Disaster Prediction, Recovery andProvision for the Future 90pages

September 30, 2011

COE Lecture Note Vol.33

若山 正人
福本 康秀
高木
剛
山本 昌宏

Study Group Workshop 2011 Lecture & Report

October 27, 2011

COE Lecture Note Vol.34

Adrian Muntean
Vladimír Chalupecký

Homogenization Method and Multiscale Modeling

COE Lecture Note Vol.35

横山 俊一
夫
紀恵
林
卓也

計算機代数システムの進展

COE Lecture Note Vol.36

Michal Beneš
Masato Kimura
Shigetoshi Yazaki

Proceedings of Czech-Japanese Seminar in Applied Mathematics 2010
107pages

COE Lecture Note Vol.37

若山 正人
高木
剛
Kirill Morozov
平岡 裕章
木村 正人
白井 朋之
西井 龍映
栄 伸一郎
穴井 宏和
福本 康秀

平成23年度 数学・数理科学と諸科学・産業との連携研究ワーク February 20, 2012
ショップ 拡がっていく数学 ～期待される “見えない力” ～154pages隆
卓
文生

157pages

128pages

140pages

72pages

210pages

October 28, 2011

November 30, 2011

January 27, 2012

シリーズ既刊
Issue

Author／Editor

Title

Published

COE Lecture Note Vol.38

Fumio Hiroshima
Itaru Sasaki
Herbert Spohn
Akito Suzuki

Enhanced Binding in Quantum Field Theory

COE Lecture Note Vol.39

Institute of Mathematics
for Industry,
Kyushu University

Multiscale Mathematics: Hierarchy of collective phenomena andinter­relations between hierarchical structures 180pages

March 13, 2012

COE Lecture Note Vol.40

井ノ口順一
太田 泰広
筧
三郎
梶原 健司
松浦
望

離散可積分系・離散微分幾何チュートリアル2012

March 15, 2012

COE Lecture Note Vol.41

Institute of Mathematics
for Industry,
Kyushu University

Forum “Math-for-Industry” 2012
“Information Recovery and Discovery”

COE Lecture Note Vol.42

佐伯
若山
山本

Study Group Workshop 2012 Abstract, Lecture & Report

178pages

November 19, 2012

COE Lecture Note Vol.43

Institute of Mathematics
for Industry,
Kyushu University

Combinatorics and Numerical Analysis Joint Workshop

103pages

December 27, 2012

COE Lecture Note Vol.44

萩原学

モダン符号理論からポストモダン符号理論への展望

107pages

January 30, 2013

COE Lecture Note Vol.45

金山

寛

Joint Research Workshop of Institute of Mathematics for Industry(IMI), Kyushu University“Propagation of Ultra-large-scale Computation by the Domaindecomposition-method for Industrial Problems (PUCDIP 2012)”
121pages

February 19, 2013

COE Lecture Note Vol.46

西井 龍映
栄 伸一郎
岡田 勘三
落合 啓之
小磯 深幸
斎藤 新悟
白井 朋之

科学・技術の研究課題への数学アプローチ―数学モデリングの基礎と展開― 325pages

February 28, 2013

COE Lecture Note Vol.47

SOO TECK LEE

BRANCHING RULES AND BRANCHING ALGEBRAS FOR THE
COMPLEX CLASSICAL GROUPS 40pages

March 8, 2013

COE Lecture Note Vol.48

溝口
脇
平坂
谷口
島袋

博多ワークショップ「組み合わせとその応用」 124pages

March 28, 2013

修
正人
昌宏

佳寛
隼人
貢
哲至
修

204pages

March 12, 2012

152pages

October 22, 2012

91pagesシリーズ既刊
Issue

Author／Editor

COE Lecture Note Vol.49

照井
小原
濱田
横山
穴井
横田

MI Lecture Note Vol.50

章
功任
龍義
俊一
宏和
博史

Title

Published

マス・フォア・インダストリ研究所 共同利用研究集会 II
数式処理研究と産学連携の新たな発展 137pages

August 9, 2013

Ken Anjyo
Hiroyuki Ochiai
Yoshinori Dobashi
Yoshihiro Mizoguchi
Shizuo Kaji

Symposium MEIS2013:
Mathematical Progress in Expressive Image Synthesis

October 21, 2013

MI Lecture Note Vol.51

Institute of Mathematics
for Industry, Kyushu
University

Forum “Math-for-Industry” 2013
“The Impact of Applications on Mathematics”

MI Lecture Note Vol.52

佐伯
岡田
髙木
若山
山本

Study Group Workshop 2013 Abstract, Lecture & Report
142pages

November 15, 2013

MI Lecture Note Vol.53

四方 義啓
櫻井 幸一
安田 貴徳
Xavier Dahan

平成25年度 九州大学マス・フォア・インダストリ研究所
共同利用研究集会 安全・安心社会基盤構築のための代数構造～サイバー社会の信頼性確保のための数理学～ 158pages

December 26, 2013

MI Lecture Note Vol.54

Takashi Takiguchi
Hiroshi Fujiwara

Inverse problems for practice, the present and the future

January 30, 2014

MI Lecture Note Vol.55

栄 伸一郎
溝口 佳寛
脇
隼人
渋田 敬史

Study Group Workshop 2013 数学協働プログラム Lecture & Report
98pages

February 10, 2014

MI Lecture Note Vol.56

Yoshihiro Mizoguchi
Hayato Waki
Takafumi Shibuta
Tetsuji Taniguchi
Osamu Shimabukuro
Makoto Tagami
Hirotake Kurihara
Shuya Chiba

Hakata Workshop 2014
~ Discrete Mathematics and its Applications ~March 28, 2014

MI Lecture Note Vol.57

Institute of Mathematics
for Industry, Kyushu
University

Forum “Math-for-Industry” 2014:“Applications + Practical Conceptualization + Mathematics = fruitful
Innovation” 93pages

October 23, 2014

MI Lecture Note Vol.58

安生健一
落合啓之

Symposium MEIS2014:
Mathematical Progress in Expressive Image Synthesis

November 12, 2014

修
勘三
剛
正人
昌宏

154pages

October 30, 2013

97pages

93pages

141pages

135pagesシリーズ既刊
Issue

Author／Editor

MI Lecture Note Vol.59

西井
岡田
梶原
髙木
若山
脇
山本

龍映
勘三
健司
剛
正人
隼人
昌宏

MI Lecture Note Vol.60

西浦博

MI Lecture Note Vol.61

Title

Published

Study Group Workshop 2014 数学協働プログラム
Abstract, Lecture & Report 196pages

November 14, 2014

平成26年度九州大学 IMI 共同利用研究・研究集会（I）
感染症数理モデルの実用化と産業及び政策での活用のための新たな展開 120pages

November 28, 2014

溝口 佳寛
Jacques Garrigue
萩原
学
Reynald Affeldt

研究集会
高信頼な理論と実装のための定理証明および定理証明器
Theorem proving and provers for reliable theory and implementations(TPP2014) 138pages

February 26, 2015

MI Lecture Note Vol.62

白井朋之

Workshop on “β-transformation and related topics”

March 10, 2015

MI Lecture Note Vol.63

白井朋之

Workshop on “Probabilistic models with determinantal structure”
107pages

August 20, 2015

MI Lecture Note Vol.64

落合
土橋

啓之
宜典

Symposium MEIS2015:
Mathematical Progress in Expressive Image Synthesis

September 18, 2015

MI Lecture Note Vol.65

Institute of Mathematics
for Industry, Kyushu
University

Forum “Math-for-Industry” 2015
“The Role and Importance of Mathematics in Innovation”

MI Lecture Note Vol.66

岡田 勘三
藤澤 克己
白井 朋之
若山 正人
脇
隼人
Philip Broadbridge
山本 昌宏

Study Group Workshop 2015 Abstract, Lecture & Report
156pages

MI Lecture Note Vol.67

Institute of Mathematics
for Industry, Kyushu
University

IMI-La Trobe Joint Conference“Mathematics for Materials Science and Processing”
66pages

February 5, 2016

MI Lecture Note Vol.68

古庄
小谷
新甫

英和
久寿
洋史

結び目と Grothendieck-Teichmüller 群
116pages

February 22, 2016

MI Lecture Note Vol.69

土橋
鍛治

宜典
静雄

Symposium MEIS2016:
Mathematical Progress in Expressive Image Synthesis

MI Lecture Note Vol.70

Institute of Mathematics
for Industry,
Kyushu University

Forum “Math-for-Industry” 2016
“Agriculture as a metaphor for creativity in all human endeavors”
98pagesNovember 2, 2016

MI Lecture Note Vol.71

小磯
二宮
山本

Study Group Workshop 2016 Abstract, Lecture & Report

November 21, 2016

深幸
嘉行
昌宏

59pages

124pages

74pages

October 23, 2015

November 5, 2015

82pages

143pages

October 24, 2016

シリーズ既刊
Issue

Author／Editor
朝雄
泉
文生

Title

Published

MI Lecture Note Vol.72

新井
小嶋
廣島

MI Lecture Note Vol.73

穴田 啓晃
Kirill Morozov
須賀 祐治
奥村 伸也
櫻井 幸一

Secret Sharing for Dependability, Usability and Security of Network
Storage and Its Mathematical Modeling 211pages

March 15, 2017

MI Lecture Note Vol.74

QUISPEL, G. Reinout W.
BADER, Philipp
MCLAREN, David I.
TAGAMI, Daisuke

IMI-La Trobe Joint Conference
Geometric Numerical Integration and its Applications

March 31, 2017

MI Lecture Note Vol.75

手塚
田上
山本

Study Group Workshop 2017 Abstract, Lecture & Report

MI Lecture Note Vol.76

宇田川誠一

Tzitzéica 方程式の有限間隙解に付随した極小曲面の構成理論―Tzitzéica 方程式の楕円関数解を出発点として― 68pages

August 4, 2017

MI Lecture Note Vol.77

松谷 茂樹
佐伯
修
中川 淳一
田上 大助
上坂 正晃
Pierluigi Cesana
濵田 裕康

平成29年度 九州大学マス・フォア・インダストリ研究所
共同利用研究集会（I）
結晶の界面，転位，構造の数理 148pages

December 20, 2017

MI Lecture Note Vol.78

瀧澤 重志
小林 和博
佐藤憲一郎
斎藤
努
清水 正明
間瀬 正啓
藤澤 克樹
神山 直之

平成29年度 九州大学マス・フォア・インダストリ研究所
プロジェクト研究 研究集会（I）
防災・避難計画の数理モデルの高度化と社会実装へ向けて
136pages

February 26, 2018

MI Lecture Note Vol.79

神山
畔上

平成29年度 AIMaP チュートリアル
最適化理論の基礎と応用 96pages

February 28, 2018

MI Lecture Note Vol.80

Kirill Morozov
Hiroaki Anada
Yuji Suga

IMI Workshop of the Joint Research Projects
Cryptographic Technologies for Securing Network Storage
and Their Mathematical Modeling 116pages

March 30, 2018

MI Lecture Note Vol.81

Tsuyoshi Takagi
Masato Wakayama
Keisuke Tanaka
Noboru Kunihiro
Kazufumi Kimoto
Yasuhiko Ikematsu

IMI Workshop of the Joint Research Projects
International Symposium on Mathematics, Quantum Theory,
and Cryptography 246pages

September 25, 2019

MI Lecture Note Vol.82

池森

令和 2 年度 AIMaP チュートリアル
新型コロナウイルス感染症にかかわる諸問題の数理
145pages

March 22, 2021

集
大助
昌宏

直之
秀幸

俊文

Mathematical quantum field theory and related topics

133pages

71pages

118pages

January 27, 2017

October 20, 2017

シリーズ既刊
Issue

Author／Editor

Title

Published

MI Lecture Note Vol.83

早川健太郎
軸丸 芳揮
横須賀洋平
可香谷 隆
林
和希
堺
雄亮

シェル理論・膜理論への微分幾何学からのアプローチとその建築曲面設計への応用 49pages

July 28, 2021

MI Lecture Note Vol.84

Taketoshi Kawabe
Yoshihiro Mizoguchi
Junichi Kako
Masakazu Mukai
Yuji Yasui

SICE-JSAE-AIMaP Tutorial
Advanced Automotive Control and Mathematics

December 27, 2021

MI Lecture Note Vol.85

Hiroaki Anada
Yasuhiko Ikematsu
Koji Nuida
Satsuya Ohata
Yuntao Wang

IMI Workshop of the Joint Usage Research Projects
February 9, 2022
Exploring Mathematical and Practical Principles of Secure Computationand Secret Sharing 114pages

MI Lecture Note Vol.86

濱田 直希
穴井 宏和
梅田 裕平
千葉 一永
佐藤 寛之
能島 裕介
加葉田雄太朗
一木 俊助
早野 健太
佐伯
修

2020年度採択分 九州大学マス・フォア・インダストリ研究所
共同利用研究集会
進化計算の数理 135pages

February 22, 2022

MI Lecture Note Vol.87

Osamu Saeki,
Ho Tu Bao,
Shizuo Kaji,
Kenji Kajiwara,
Nguyen Ha Nam,
Ta Hai Tung,
Melanie Roberts,
Masato Wakayama,
Le Minh Ha,
Philip Broadbridge

Proceedings of Forum “Math-for-Industry” 2021
-Mathematics for Digital Economy- 122pages

March 28, 2022

MI Lecture Note Vol.88

Daniel PACKWOOD
Pierluigi CESANA,
Shigenori FUJIKAWA,
Yasuhide FUKUMOTO,
Petros SOFRONIS,Alex STAYKOVPerspectives on Artificial Intelligence and Machine Learning inMaterials Science, February 4-6, 2022 74pages

November 8, 2022

110pagesシリーズ既刊
Issue

Author／Editor

Title

Published

MI Lecture Note Vol.89

松谷 茂樹
落合 啓之
井上 和俊
小磯 深幸
佐伯
修
白井 朋之
垂水 竜一
内藤 久資
中川 淳一
濵田 裕康
松江
要
加葉田雄太朗2022年度採択分 九州大学マス・フォア・インダストリ研究所
共同利用研究集会
材料科学における幾何と代数 III 356pages

December 7, 2022

MI Lecture Note Vol.90

中山
谷川
品野
近藤
石原
鍛冶
藤澤

尚子
拓司
勇治
正章
亨
静雄
克樹

2022年度採択分 九州大学マス・フォア・インダストリ研究所
共同利用研究集会
データ格付けサービス実現のための数理基盤の構築 58pages

December 12, 2022

MI Lecture Note Vol.91

Katsuki Fujisawa
Shizuo Kaji
Toru Ishihara
Masaaki Kondo
Yuji Shinano
Takuji Tanigawa
Naoko Nakayama

IMI Workshop of the Joint Usage Research Projects
Construction of Mathematical Basis for Realizing Data Rating Service
610pages

December 27, 2022

MI Lecture Note Vol.92

丹田
三宮
廣島

2022年度採択分 九州大学マス・フォア・インダストリ研究所
共同利用研究集会
時間・量子測定・準古典近似の理論と実験〜古典論と量子論の境界〜 150pages

Janualy 6, 2023

MI Lecture Note Vol.93

Philip Broadbridge
Luke Bennetts
Melanie Roberts
Kenji Kajiwara

Proceedings of Forum “Math-for-Industry” 2022
-Mathematics of Public Health and Sustainability-

June 19, 2023

MI Lecture Note Vol.94

國廣
池松
伊豆
穴田
縫田

2023年度採択分 九州大学マス・フォア・インダストリ研究所
共同利用研究集会
現代暗号に対する安全性解析・攻撃の数理 260pages

聡
俊
文生

昇
泰彦
哲也
啓晃
光司

170pages

Janualy 11, 2024

九州大学マス・フォア・インダストリ研究所

九州大学大学院 数理学府
URL http://www.imi.kyushu-u.ac.jp/