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Dynamical invariants and measures on metric graphs and Applications in medical science : 거리그래프의 동역학적 불변량 및 측도와 의학 분야의 응용

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dc.contributor.advisor임선희-
dc.contributor.author박재민-
dc.date.accessioned2023-06-29T02:35:46Z-
dc.date.available2023-06-29T02:35:46Z-
dc.date.issued2023-
dc.identifier.other000000176210-
dc.identifier.urihttps://hdl.handle.net/10371/194354-
dc.identifier.urihttps://dcollection.snu.ac.kr/common/orgView/000000176210ko_KR
dc.description학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2023. 2. 임선희.-
dc.description.abstractThe space of geodesics on a metric graph has three important invariant measures for geodesic flow that reflect the geometric, dynamical, and probabilistic properties of the metric graph. The measures are constructed by dynamical invariants and measure classes on the boundary of the universal covering tree. In this thesis, we focus on the structure of the metric graphs that determines the dynamical invariants and the boundary measure classes.

First, we formulate three boundary measure classes using potential functions analogous to the manifold cases: visibility measures, Patterson-Sullivan measures, and harmonic measures. We show that there is an edge length which is a necessary and sufficient condition to the equivalence of two of these measure classes .

Next, we use the dynamical invariant and boundary measures to study the brain network. Regarding the brain network as a metric graph, we compute the volume entropy and Patterson-Sullivan measure numerically. Comparing the values between the tinnitus group and the non-tinnitus group, we strengthen the tinnitus cause interpretation based on the Bayesian hypothesis and the triple network model.

We also obtain a result of topological data analysis on medical science. Using the Mapper algorithm, we represent data space as a metric graph and propose a grouping method based on the structure of the metric graph. In this framework, we find the new subtype of Mitral regurgitation patients.

Finally, we improve a well-known result in the Diophantine approximation. We construct a fractal set contained in weighted singular vectors using tree structure and the shadowing property in homogeneous dynamics. By constructing the tree associated to lattice point counting, we obtain a nontrivial lower bound of Hausdorff dimension of weighted singular vectors.		-
dc.description.abstract거리그래프의 측지선 공간에는 거리 그래프의 기하학적, 동역학적 및 확률론적 특징을 반영하는 측지적 류에 대한 세 가지 중요한 불변측도가 있다. 측도는 동역학적 불변량과 범피복나무의 경계에서 정의된 측도류에 의해 구축된다. 본 학위논문에서는 동역학적 불변량과 경계 측도류를 결정하는 거리그래프의 구조에 집중한다.

먼저 다양체의 경우와 유사하게 퍼텐셜함수를 이용해 비지빌리티 측도, 패터슨-설리반 측도, 하모닉 측도, 총 세 가지 경측도를 공식화한다. 이러한 경측도류 중 두 개가 동치일 필요충분조건이 특정한 간선 길이에 대한 조건으로 나타남을 보인다.

다음은 동역학적 불변량과 경계 측도를 활용한 뇌 네트워크 연구이다. 뇌 네트워크를 거리그래프로 간주하여 부피 엔트로피와 패터슨-설리반 측도를 수치적으로 계산한다. 이명 집단과 비이명 집단에서 이 값들을 비교하여 베이지안 가설에 기반해 이명 증상의 원인을 해석한다.

또한 의료수학에서 위상수학적 데이터 분석에 대한 결과를 소개한다. 매퍼 알고리즘을 이용해 데이터 공간을 거리그래프로 나타내고 거리그래프의 구조에 기반한 그룹화 방법을 제안한다. 이러한 방법론에 기반해 승모판막 협착증 환자들의 새로운 하위 유형을 찾는다.

마지막으로 디오판틴 근사 분야의 결과를 향상시킨 결과를 소개한다. 나무의 구조와 균질 동역학의 투영 성질을 이용해 무게 특이 벡터들 안에 속하는 프랙탈 집합을 만든다. 격자점 셈을 통해 나무를 관찰해 무게 특이 벡터들의 하우스도르프 차원의 하계를 얻는다.		-
dc.description.tableofcontents1 Introduction 1
1.1 Equivalence of boundary measures 2
1.2 Brain network analysis 1: Tinnitus on hearing loss patients 5
1.3 Brain network analysis 2: Tinnitus on sudden sensorineural hearing loss patients 7
1.4 Topological data analysis: Mitral regurgitation 9
1.5 Lower bound of Hausdorff dimension of weighted singular vectors 9
2 Dynamics on Metric graphs 13
2.1 Geometry on metric graphs 13
2.1.1 Metric graphs 13
2.1.2 Spaces of geodesics 14
2.2 Potentials, Critical exponents, and Gibbs cocycles 16
2.2.1 Potentials 16
2.2.2 Critical exponents and Gibbs cocycles 17
2.2.3 Systems of conductances 18
2.3 Gibbs measures 18
2.3.1 Patterson densities 19
2.3.2 Gibbs measures 22
2.4 Variational principles 23
2.4.1 Entropy 23
2.4.2 Topological Pressure 24
2.5 Coding 25
2.5.1 Two-sided subshifts of finite type 26
2.5.2 Coding discrete-time geodesic flows 26
2.5.3 Suspensions 28
2.5.4 Coding continuous-time geodesic flows 29
3 Equivalence of boundary measures 31
3.1 Visibility measures 33
3.2 Patterson-Sullivan measures 38
3.3 Harmonic measures 39
3.4 Equivalence conditions 44
4 Brain network analysis 1: Tinnitus on hearing loss patients 57
4.1 Materials and Methods 57
4.1.1 Patients 57
4.1.2 EEG recording 59
4.1.3 Source localization analysis 59
4.1.4 Metric graph 60
4.1.5 Volume entropy 60
4.1.6 Afferent node capacity 62
4.1.7 Statistical analysis 63
4.2 Results 64
4.2.1 Comparison of volume entropy between the HL-T and HL-NT groups 64
4.2.2 Comparison of afferent node capacity between the HL-T and HL-NT groups 64
4.3 Discussion 67
4.4 Conclusion 72
5 Brain network analysis 2: Tinnitus on sudden sensorineural hearing loss patients 75
5.1 Materials and Methods 75
5.1.1 Participants 75
5.1.2 Electroencephalography recording 76
5.2 Results 78
5.2.1 Comparison of the volume entropy between the sudden sensorineural hearing loss-with tinnitus and sudden sensorineural hearing loss-without tinnitus groups 78
5.2.2 Comparison of a erent node capacity between the sudden sensorineural hearing loss-with tinnitus and sudden sensorineural hearing loss-without tinnitus groups 78
5.3 Discussion 80
5.3.1 New insight into the generation of tinnitus in patients with sudden sensorineural hearing loss provided by a triple network model 80
5.3.2 Activation of auditory processing and noise-canceling pathways in sudden sensorineural hearing loss patients without tinnitus 82
5.3.3 Study strengths and limitations 83
5.4 Conclusion 85
6 Topological data analysis: Mitral regurgitation 88
6.1 Methods 88
6.1.1 Study participants 88
6.1.2 Echocardiographic evaluation 89
6.1.3 Topological data analysis 91
6.1.4 Phenogrouping based on patient-patient similarity network model 91
6.1.5 Clinical outcomes 94
6.1.6 Statistical analysis 94
6.2 Results 94
6.2.1 Study population of the derivation cohort 94
6.2.2 Patient-patient similarity network model and distinct phenogroups of primary MR patients 95
6.2.3 Characteristics of primary MR phenogroups 95
6.2.4 Association of the distinct phenotypic groups with clinical outcome and its additive value 99
6.2.5 Validation of network model of primary MR 99
6.3 Discussion 109
6.3.1 Study limitation 111
6.4 Conclusion 111
6.5 Supplemental Methods 112
6.5.1 Echocardiographic evaluation 112
6.5.2 Mapper 112
6.5.3 Grouping 116
7 Lower bound of Hausdorff dimension of weighted singular vectors 119
7.1 Fractal structure and Hausdorff dimension 119
7.1.1 Fractal structure 119
7.1.2 Self-affine structure and lower bound 120
7.2 Counting lattice points in convex sets 129
7.2.1 Preliminaries for lattice point counting 129
7.2.2 Lattice point counting in $\mathbb{R}^{d+1}$ 132
7.3 Lower bound 142
7.3.1 Construction of the fractal set 142
7.3.2 The lower bound calculation 148
Abstract (in Korean) 174
Acknowledgement (in Korean) 175		-
dc.format.extentvi, 173-
dc.language.isoeng-
dc.publisher서울대학교 대학원-
dc.subjectmetric graph-
dc.subjectdynamical invariant-
dc.subjectinvariant measure-
dc.subjectbrain network-
dc.subjecttopological data analysis-
dc.subjectDiophantine approximation-
dc.subject.ddc510-
dc.titleDynamical invariants and measures on metric graphs and Applications in medical science-
dc.title.alternative거리그래프의 동역학적 불변량 및 측도와 의학 분야의 응용-
dc.typeThesis-
dc.typeDissertation-
dc.contributor.AlternativeAuthorJaemin Park-
dc.contributor.department자연과학대학 수리과학부-
dc.description.degree박사-
dc.date.awarded2023-02-
dc.identifier.uciI804:11032-000000176210-
dc.identifier.holdings000000000049▲000000000056▲000000176210▲-
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