5 research outputs found
๋จ์ฒด๋ณตํฉ์ฒด์์ ์กฐํ ๊ณต๊ฐ์ ์ด๋ก ๊ณผ ์์ฉ
ํ์๋
ผ๋ฌธ(๋ฐ์ฌ)--์์ธ๋ํ๊ต ๋ํ์ :์์ฐ๊ณผํ๋ํ ์๋ฆฌ๊ณผํ๋ถ,2020. 2. ๊ตญ์
.A harmonic cycle ฮป, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a simplicial chain complex. By combinatorial Hodge theory, harmonic spaces are isomorphic to homology groups with real coefficients. In particular, if a cell complex has a reduced homology with Betti number ฮฒ_i = 1 of a specific dimension i, it has a unique harmonic cycle up to scalar multiplication, which we call the standard harmonic cycle.
We will present a formula for the standard harmonic cycle ฮป of a cell complex based on a high-dimensional generalization of cycletrees. Moreover, by using duality, we will define the standard harmonic cocycle ฮป* and show intriguing combinatorial properties of ฮป and ฮป* in relation to (dual) spanning trees, (dual) cycletrees, winding numbers w( ยท ) and cutting numbers c( ยท ) in high dimensions.
Finally, we will also suggest two application methods; an analysis to detect oscillations by using winding number, and cutting number, and a network embedding method, called harmonic mirroring.์กฐํ ์ฌ์ดํด ฮป๋ ์ด์ฐ ์กฐํ ํ์์ผ๋ก๋ ๋ถ๋ฅด๋ฉฐ ๋ผํ๋ผ์์ ๋ฐฉ์ ์์ ํด์ด๋ค. ์ด ๋ผํ๋ผ์์ ๋ฐฉ์ ์์ ๋จ์ฒด ์ฐ์๋ณตํฉ์ฒด์ ๊ฒฝ๊ณ ์์ฉ์๋ก๋ถํฐ ๋ง๋ ์กฐํฉ๋ก ์ ๋ผํ๋ผ์์ ์์ฉ์๊ฐ 0์ผ ๋ ์๊ธฐ๋ ์์์ด๋ค. ์กฐํฉ๋ก ์ ํธ์ง ์ด๋ก ์ ์ํ์ฌ ์กฐํ ๊ณต๊ฐ์ ์ค์๋ฅผ ๊ณ์๋ก ๊ฐ๋ ํธ๋ชฐ๋ก์ง๊ตฐ๊ณผ ๋ํ์ด๋ค. ํนํ ์ฐ์ ๋ณตํฉ์ฒด์ ํน์ ์ฐจ์ i์์ ์ถ์ ํธ๋ชฐ๋ฆฌ์ง๊ตฐ์ ๋ฒ ํฐ์ ฮฒ_i๊ฐ 1์ด๋ผ๋ฉด, ์ค์นผ๋ผ๊ณฑ์ ์ ์ธํ๊ณ ๋ถ๋ณํ๋ ๊ณ ์ ํ ์กฐํ ์ฌ์ดํด์ ์ป์ ์ ์๊ณ , ์ด๋ฅผ ํ์ค ์กฐํ ์ฌ์ดํด์ด๋ผ ๋ถ๋ฅธ๋ค. ์ฐ๋ฆฌ๋ ํ์ค ์กฐํ ์ฌ์ดํด ฮป์ ํํํ๋ ๊ณต์์ ๊ณ ์ฐจ์์ผ๋ก ์ผ๋ฐํ๋ ์ฌ์ดํด ํธ๋ฆฌ๋ฅผ ๋ฐํ์ผ๋ก ๋ํ๋ด์๋ค. ๋์ฑ์ด, ์๋์ฑ์ ์ด์ฉํ์ฌ ํ์ค ์กฐํ ์๋์ฌ์ดํด ฮป*๋ฅผ ์ ์ํ์๊ณ , ฮป์ ฮป*์ ํฅ๋ฏธ๋กญ๊ณ ์กฐํฉ๋ก ์ ์ธ ์ฑ์ง๋ค์ ๊ณ ์ฐจ์ ์ํ์์ ๋ณด์๋ค. ์ด๋ (์๋) ์์ฑ๋๋ฌด์ (์๋) ์ฌ์ดํด ํธ๋ฆฌ, ํ์ ์ w( ยท ), ์๋ฆ์ c( ยท )์์ ๊ด๊ณ๋ฅผ ๊ฐ๋๋ค. ๋ง์ง๋ง์ผ๋ก ์ฐ๋ฆฌ๋ ์์ฉ์ ์ํ ๋ ๊ฐ์ง ๋ฐฉ๋ฒ๋ก ์ ์ ์ํ๋ค. ํ์ ์์ ์๋ฆ์๋ฅผ ์ด์ฉํ ์ง๋ ์ธก์ ๋ฒ๊ณผ ์กฐํ ๋ฏธ๋ฌ๋ง์ผ๋ก ๋ถ๋ฆฌ๋ ๋คํธ์ํฌ ๋งค์ฅ ๋ฐฉ๋ฒ์ด๋ค.1 Introduction 1
2 Preliminaries 3
2.1 Review of nite chain complex and (co)homology 3
2.2 High dimensional spanning trees 4
2.3 Harmonic space and combinatorial Hodge theory 5
3 Cycletree and its minimal cycle 7
3.1 Cycletree 7
3.2 Minimal cycle 9
4 Winding number 14
5 Standard harmonic cycle 17
6 Duality and dual spanning tree 20
7 Dual cycletree and cutting number 25
7.1 Dual cycletree and its minimal cocycle 25
7.2 Cutting number 27
8 Standard harmonic cocycle and relationship 31
9 Application 35
9.1 Oscillation Detection 35
9.2 Harmonic Mirroring 38
Abstract (in Korean) 41
Acknowledgement (in Korean) 42Docto
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