"호몰로지"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(같은 사용자의 중간 판 5개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
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* simplicial 호몰로지(코호몰로지)
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* [[단체 호몰로지 (simplicial homology)]]
−
* singular 호몰로지
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* 특이 호몰로지 (singular homology)
−
* [[드람 코호몰로지]]
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* <math>\langle S,d\omega\rangle=\langle \partial S,\omega \rangle</math>
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* Eilenberg-Steenrod 공리
+
 
 +
 
 +
==에일렌베르크-스틴로드 (Eilenberg-Steenrod) 공리==
 +
* '호몰로지 이론은 에일렌베르크-스틴로드 공리를 만족하는 functor이다'
  
−
* <math>\langle S,d\omega\rangle=\langle \partial S,\omega \rangle</math>
 
  
  
39번째 줄: 41번째 줄:
  
 
==사전 형태의 자료==
 
==사전 형태의 자료==
−
* http://en.wikipedia.org/wiki/Simplicial_homology
+
* http://en.wikipedia.org/wiki/Homology_theory
−
 
 
 
   
 
   
  
 
==관련논문==
 
==관련논문==
 +
* Weibel, Charles A. 1999. [http://www.math.uiuc.edu/K-theory/0245/ History of Homological Algebra] History of Topology: 797–836.
 +
* Hilton, Peter [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century] (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291
 +
 +
== 노트 ==
 +
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q1144780 Q1144780]
 +
===말뭉치===
 +
# Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf.<ref name="ref_ad36c7c4">[https://encyclopediaofmath.org/wiki/Homology_theory Encyclopedia of Mathematics]</ref>
 +
# Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory.<ref name="ref_ad36c7c4" />
 +
# There exists a cohomology theory dual to a homology theory (cf.<ref name="ref_ad36c7c4" />
 +
# The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms.<ref name="ref_ad36c7c4" />
 +
# By analysis of the lifting problem it introduces the funda­ mental group and explores its properties, including Van Kampen's Theorem and the relationship with the first homology group.<ref name="ref_2661a95e">[https://www.springer.com/gp/book/9780387941264 Homology Theory - An Introduction to Algebraic Topology]</ref>
 +
# Conceptually, however, it can be useful to understand homology as a special kind of homotopy.<ref name="ref_8773a6e8">[https://ncatlab.org/nlab/show/homology homology in nLab]</ref>
 +
# This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts.<ref name="ref_8773a6e8" />
 +
# One good way of understanding homology of CW complexes is with cellular homology.<ref name="ref_5f73e239">[https://math.stackexchange.com/questions/40149/intuition-of-the-meaning-of-homology-groups Intuition of the meaning of homology groups]</ref>
 +
# Homology groups were originally defined in algebraic topology .<ref name="ref_4f6d6ab8">[https://en.wikipedia.org/wiki/Homology_(mathematics) Homology (mathematics)]</ref>
 +
# The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.<ref name="ref_4f6d6ab8" />
 +
# Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold .<ref name="ref_4f6d6ab8" />
 +
# A particular type of mathematical object, such as a topological space or a group , may have one or more associated homology theories.<ref name="ref_4f6d6ab8" />
 +
# Historically, the term "homology" was first used in a topological sense by Poincaré.<ref name="ref_53ee2d6c">[https://mathworld.wolfram.com/Homology.html Homology -- from Wolfram MathWorld]</ref>
 +
# To him, it meant pretty much what is now called a bordism, meaning that a homology was thought of as a relation between manifolds mapped into a manifold.<ref name="ref_53ee2d6c" />
 +
# To simplify the definition of homology, Poincaré simplified the spaces he dealt with.<ref name="ref_53ee2d6c" />
 +
# Eventually, Poincaré's version of homology was dispensed with and replaced by the more general singular homology.<ref name="ref_53ee2d6c" />
 +
# The starting point will be simplicial complexes and simplicial homology.<ref name="ref_c378c1ed">[https://warwick.ac.uk/fac/sci/maths/undergrad/ughandbook/year3/ma3h6/ MA3H6 Algebraic Topology]</ref>
 +
# The simplicial homology depends on the way these simplices fit together to form the given space.<ref name="ref_c378c1ed" />
 +
# It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition.<ref name="ref_c378c1ed" />
 +
# One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes.<ref name="ref_c378c1ed" />
 +
# We can think of homology as a perspective that allows one to distinguish shapes and forms disregarding changes made by stretching and bending.<ref name="ref_c6801dd4">[https://www.maths.ox.ac.uk/about-us/life-oxford-mathematics/oxford-mathematics-alphabet/h-homology Mathematical Institute]</ref>
 +
# For example, homology sees no difference between the beach ball and the jumping ball, which can be constructed from the ball shape by pulling out two fingers.<ref name="ref_c6801dd4" />
 +
# This is the reason why according to homology those objects are not the same.<ref name="ref_c6801dd4" />
 +
# Homology is a mathematical way of counting different types of loops and holes in topological spaces.<ref name="ref_c6801dd4" />
 +
# Through analysis of viral and simulated genomic datasets, we show how persistent homology captures fundamental evolutionary aspects not directly inferred from phylogeny.<ref name="ref_a24f2602">[https://www.pnas.org/content/110/46/18566 Topology of viral evolution]</ref>
 +
# We can define a topological invariant called the “homology group” H k as an algebraic structure that encompasses all holes in dimension k, and the “Betti number” b k is the count of these holes.<ref name="ref_a24f2602" />
 +
# C. We then perform persistent homology, which computes the homology groups of dimension k at all scales ε.<ref name="ref_a24f2602" />
 +
# Our aim, then, is to apply persistent homology to the study of evolution.<ref name="ref_a24f2602" />
 +
# Exercise Show that E(V) has zero homology.<ref name="ref_124a4c9e">[https://www.win.tue.nl/~aeb/at/algtop-6.html Algebraic Topology: Homology]</ref>
 +
# For each cover we obtain a homology group.<ref name="ref_124a4c9e" />
 +
# This leads to a homomorphism of homology groups.<ref name="ref_124a4c9e" />
 +
# Let us look at .Cech homology again.<ref name="ref_124a4c9e" />
 +
# In this section, we categorize the persistent homology enabled applications as single graph and multiple graph analysis.<ref name="ref_8fadc1af">[https://appliednetsci.springeropen.com/articles/10.1007/s41109-019-0179-3 Persistence homology of networks: methods and applications]</ref>
 +
# In some applications, persistent homology is used to detect global structural features of a single network such as complexity and distributions of strongly connected regions.<ref name="ref_8fadc1af" />
 +
# (2018) use persistent homology to detect clique communities and their evolution in weighted networks.<ref name="ref_8fadc1af" />
 +
# Persistent homology is also used to analyze the brain networks by computing distributions of cliques (brain regions) and cycles (strongly connected regions) in them.<ref name="ref_8fadc1af" />
 +
# If I is an ideal of R, he considers the homology of the kernel of F * → F * /I and shows that it is independent of the choice of resolution.<ref name="ref_ffe3293e">[https://www.sciencedirect.com/topics/mathematics/homology-group Homology Group - an overview]</ref>
 +
# At first, calculations of group homology were restricted to those groups π which were fundamental groups of familiar topological spaces, using the bar complex.<ref name="ref_ffe3293e" />
 +
# By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated.<ref name="ref_5cf3f9f5">[https://www.frontiersin.org/articles/10.3389/fpls.2018.00553/full Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace]</ref>
 +
# Landmark analysis excels in its interpretability, because each landmark is an identifiable feature with biological meaning imparted by the shared homology between samples.<ref name="ref_5cf3f9f5" />
 +
# Here, we present a morphometric technique based on topology, using a persistent homology framework, to measure the outlines of leaves and classify them by plant family.<ref name="ref_5cf3f9f5" />
 +
# Using persistent homology, we then use a linear discriminant analysis (LDA) to classify leaves by plant family.<ref name="ref_5cf3f9f5" />
 +
# The theory has applications in many branches of mathematics, including spectral theory, the theory of de Rham homology in differential geometry, automatic continuity theory and K-theory.<ref name="ref_0dfe9e8f">[https://www.ncl.ac.uk/maths-physics/research/pure/topological-homology/ Mathematics, Statistics and Physics, School of]</ref>
 +
# Historically, the term ``homology'' was first used in a topological sense by Poincaré .<ref name="ref_985a0d8e">[https://archive.lib.msu.edu/crcmath/math/math/h/h342.htm Homology (Topology)]</ref>
 +
# To him, it meant pretty much what is now called a Cobordism, meaning that a homology was thought of as a relation between Manifolds mapped into a Manifold.<ref name="ref_985a0d8e" />
 +
# Eventually, Poincaré's version of homology was dispensed with and replaced by the more general Singular Homology.<ref name="ref_985a0d8e" />
 +
# In modern usage, however, the word homology is used to mean Homology Group.<ref name="ref_985a0d8e" />
 +
# This workshop will explore topics of current interest in the theory of Floer homology for 3-manifolds.<ref name="ref_3da8e5f0">[http://scgp.stonybrook.edu/archives/28596 Floer homology in low-dimensional topology: January 11-15, 2021]</ref>
 +
# Floer homology is a powerful tool for studying the topology of 3- and 4-dimensional manifolds, and the relations between them.<ref name="ref_3da8e5f0" />
 +
# There are a wide variety of ways to define the Floer homology of a 3-manifold.<ref name="ref_3da8e5f0" />
 +
# Many of these are now known to be equivalent, but their relation to the original instanton homology remains mysterious.<ref name="ref_3da8e5f0" />
 +
# In the Theory section, we have shown that the number of critical simplices determines the effective number of filtration weights to study the persistent homology of a clique complex (See Eq. 10).<ref name="ref_d8d23b01">[https://www.nature.com/articles/s41598-019-50202-3 Persistent homology of unweighted complex networks via discrete Morse theory]</ref>
 +
# Rather, our main goal is to develop a systematic filtration scheme to study persistent homology in unweighted and undirected networks.<ref name="ref_d8d23b01" />
 +
# A visual inspection of the barcode diagrams for the five model networks (Figs 3 and 5 and SI Figs S1–S4) suggests that the different models can be distinguished based on their persistent homology.<ref name="ref_d8d23b01" />
 +
# The last observation is a reflection of the differences in the persistent homology of the clique complexes corresponding to spherical and hyperbolic networks.<ref name="ref_d8d23b01" />
 +
# In this paper, we propose to use the persistent homology to systematically study the osmolytes’ molecular aggregation and their hydrogen-bonding network from a global topological perspective.<ref name="ref_b42e2c02">[https://pubs.rsc.org/en/content/articlelanding/2019/cp/c9cp03009c Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks]</ref>
 +
# The -th homology group of a simplicial complex , denoted , is the quotient vector space .<ref name="ref_0711f2e7">[https://jeremykun.com/2013/04/03/homology-theory-a-primer/ Homology Theory — A Primer]</ref>
 +
# Therefore, computing homology generalizes the graph-theoretic methods of computing connected components.<ref name="ref_0711f2e7" />
 +
# The quotient construction imposes that two vertices (two elements of the homology group) are considered equivalent if their difference is a boundary.<ref name="ref_0711f2e7" />
 +
# It is easy to see that (indeed, just by the first four generators of the image) all vertices are equivalent to 0, so there is a unique generator of homology, and the vector space is isomorphic to .<ref name="ref_0711f2e7" />
 +
===소스===
 +
<references />
  
−
* [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century]<br>
+
==메타데이터==
−
** Hilton, Peter (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291
+
===위키데이터===
−
* http://www.jstor.org/action/doBasicSearch?Query=
+
* ID :  [https://www.wikidata.org/wiki/Q1144780 Q1144780]
−
* http://www.ams.org/mathscinet
+
===Spacy 패턴 목록===
−
* http://dx.doi.org/
+
* [{'LEMMA': 'homology'}]

2021년 2월 17일 (수) 02:51 기준 최신판

개요


에일렌베르크-스틴로드 (Eilenberg-Steenrod) 공리

  • '호몰로지 이론은 에일렌베르크-스틴로드 공리를 만족하는 functor이다'


역사


메모


관련된 항목들


계산 리소스


사전 형태의 자료


관련논문

노트

위키데이터

말뭉치

  1. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf.[1]
  2. Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory.[1]
  3. There exists a cohomology theory dual to a homology theory (cf.[1]
  4. The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms.[1]
  5. By analysis of the lifting problem it introduces the funda­ mental group and explores its properties, including Van Kampen's Theorem and the relationship with the first homology group.[2]
  6. Conceptually, however, it can be useful to understand homology as a special kind of homotopy.[3]
  7. This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts.[3]
  8. One good way of understanding homology of CW complexes is with cellular homology.[4]
  9. Homology groups were originally defined in algebraic topology .[5]
  10. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.[5]
  11. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold .[5]
  12. A particular type of mathematical object, such as a topological space or a group , may have one or more associated homology theories.[5]
  13. Historically, the term "homology" was first used in a topological sense by Poincaré.[6]
  14. To him, it meant pretty much what is now called a bordism, meaning that a homology was thought of as a relation between manifolds mapped into a manifold.[6]
  15. To simplify the definition of homology, Poincaré simplified the spaces he dealt with.[6]
  16. Eventually, Poincaré's version of homology was dispensed with and replaced by the more general singular homology.[6]
  17. The starting point will be simplicial complexes and simplicial homology.[7]
  18. The simplicial homology depends on the way these simplices fit together to form the given space.[7]
  19. It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition.[7]
  20. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes.[7]
  21. We can think of homology as a perspective that allows one to distinguish shapes and forms disregarding changes made by stretching and bending.[8]
  22. For example, homology sees no difference between the beach ball and the jumping ball, which can be constructed from the ball shape by pulling out two fingers.[8]
  23. This is the reason why according to homology those objects are not the same.[8]
  24. Homology is a mathematical way of counting different types of loops and holes in topological spaces.[8]
  25. Through analysis of viral and simulated genomic datasets, we show how persistent homology captures fundamental evolutionary aspects not directly inferred from phylogeny.[9]
  26. We can define a topological invariant called the “homology group” H k as an algebraic structure that encompasses all holes in dimension k, and the “Betti number” b k is the count of these holes.[9]
  27. C. We then perform persistent homology, which computes the homology groups of dimension k at all scales ε.[9]
  28. Our aim, then, is to apply persistent homology to the study of evolution.[9]
  29. Exercise Show that E(V) has zero homology.[10]
  30. For each cover we obtain a homology group.[10]
  31. This leads to a homomorphism of homology groups.[10]
  32. Let us look at .Cech homology again.[10]
  33. In this section, we categorize the persistent homology enabled applications as single graph and multiple graph analysis.[11]
  34. In some applications, persistent homology is used to detect global structural features of a single network such as complexity and distributions of strongly connected regions.[11]
  35. (2018) use persistent homology to detect clique communities and their evolution in weighted networks.[11]
  36. Persistent homology is also used to analyze the brain networks by computing distributions of cliques (brain regions) and cycles (strongly connected regions) in them.[11]
  37. If I is an ideal of R, he considers the homology of the kernel of F * → F * /I and shows that it is independent of the choice of resolution.[12]
  38. At first, calculations of group homology were restricted to those groups π which were fundamental groups of familiar topological spaces, using the bar complex.[12]
  39. By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated.[13]
  40. Landmark analysis excels in its interpretability, because each landmark is an identifiable feature with biological meaning imparted by the shared homology between samples.[13]
  41. Here, we present a morphometric technique based on topology, using a persistent homology framework, to measure the outlines of leaves and classify them by plant family.[13]
  42. Using persistent homology, we then use a linear discriminant analysis (LDA) to classify leaves by plant family.[13]
  43. The theory has applications in many branches of mathematics, including spectral theory, the theory of de Rham homology in differential geometry, automatic continuity theory and K-theory.[14]
  44. Historically, the term ``homology was first used in a topological sense by Poincaré .[15]
  45. To him, it meant pretty much what is now called a Cobordism, meaning that a homology was thought of as a relation between Manifolds mapped into a Manifold.[15]
  46. Eventually, Poincaré's version of homology was dispensed with and replaced by the more general Singular Homology.[15]
  47. In modern usage, however, the word homology is used to mean Homology Group.[15]
  48. This workshop will explore topics of current interest in the theory of Floer homology for 3-manifolds.[16]
  49. Floer homology is a powerful tool for studying the topology of 3- and 4-dimensional manifolds, and the relations between them.[16]
  50. There are a wide variety of ways to define the Floer homology of a 3-manifold.[16]
  51. Many of these are now known to be equivalent, but their relation to the original instanton homology remains mysterious.[16]
  52. In the Theory section, we have shown that the number of critical simplices determines the effective number of filtration weights to study the persistent homology of a clique complex (See Eq. 10).[17]
  53. Rather, our main goal is to develop a systematic filtration scheme to study persistent homology in unweighted and undirected networks.[17]
  54. A visual inspection of the barcode diagrams for the five model networks (Figs 3 and 5 and SI Figs S1–S4) suggests that the different models can be distinguished based on their persistent homology.[17]
  55. The last observation is a reflection of the differences in the persistent homology of the clique complexes corresponding to spherical and hyperbolic networks.[17]
  56. In this paper, we propose to use the persistent homology to systematically study the osmolytes’ molecular aggregation and their hydrogen-bonding network from a global topological perspective.[18]
  57. The -th homology group of a simplicial complex , denoted , is the quotient vector space .[19]
  58. Therefore, computing homology generalizes the graph-theoretic methods of computing connected components.[19]
  59. The quotient construction imposes that two vertices (two elements of the homology group) are considered equivalent if their difference is a boundary.[19]
  60. It is easy to see that (indeed, just by the first four generators of the image) all vertices are equivalent to 0, so there is a unique generator of homology, and the vector space is isomorphic to .[19]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Encyclopedia of Mathematics
  2. ↑ Homology Theory - An Introduction to Algebraic Topology
  3. ↑ 3.0 3.1 homology in nLab
  4. ↑ Intuition of the meaning of homology groups
  5. ↑ 5.0 5.1 5.2 5.3 Homology (mathematics)
  6. ↑ 6.0 6.1 6.2 6.3 Homology -- from Wolfram MathWorld
  7. ↑ 7.0 7.1 7.2 7.3 MA3H6 Algebraic Topology
  8. ↑ 8.0 8.1 8.2 8.3 Mathematical Institute
  9. ↑ 9.0 9.1 9.2 9.3 Topology of viral evolution
  10. ↑ 10.0 10.1 10.2 10.3 Algebraic Topology: Homology
  11. ↑ 11.0 11.1 11.2 11.3 Persistence homology of networks: methods and applications
  12. ↑ 12.0 12.1 Homology Group - an overview
  13. ↑ 13.0 13.1 13.2 13.3 Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace
  14. ↑ Mathematics, Statistics and Physics, School of
  15. ↑ 15.0 15.1 15.2 15.3 Homology (Topology)
  16. ↑ 16.0 16.1 16.2 16.3 Floer homology in low-dimensional topology: January 11-15, 2021
  17. ↑ 17.0 17.1 17.2 17.3 Persistent homology of unweighted complex networks via discrete Morse theory
  18. ↑ Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks
  19. ↑ 19.0 19.1 19.2 19.3 Homology Theory — A Primer

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LEMMA': 'homology'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=호몰로지&oldid=52077"