"다양체"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
(→‎노트: 새 문단)
 
11번째 줄: 11번째 줄:
 
* James, I. M., ed. 1999. History of Topology. Amsterdam: North-Holland. http://www.ams.org/mathscinet-getitem?mr=1674906.
 
* James, I. M., ed. 1999. History of Topology. Amsterdam: North-Holland. http://www.ams.org/mathscinet-getitem?mr=1674906.
 
** Scholz, Erhard. 1999. “The Concept of Manifold, 1850–1950.” In History of Topology, 25–64. Amsterdam: North-Holland.
 
** Scholz, Erhard. 1999. “The Concept of Manifold, 1850–1950.” In History of Topology, 25–64. Amsterdam: North-Holland.
 +
 +
== 노트 ==
 +
 +
* Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance.<ref name="ref_73fe">[https://en.wikipedia.org/wiki/History_of_manifolds_and_varieties History of manifolds and varieties]</ref>
 +
* In the same vein, the Japanese word "多様体" (tayōtai) also encompasses both manifold and variety.<ref name="ref_73fe" />
 +
* The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness".<ref name="ref_73fe" />
 +
* Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold.<ref name="ref_73fe" />
 +
* The way these connect to one another dictates the control options of a manifold.<ref name="ref_edc5">[https://www.denleyhydraulics.co.uk/news/the-benefits-and-uses-of-manifold-blocks/ The Benefits and Uses of Manifold Blocks]</ref>
 +
* A Drilled manifold, on the other hand, is made with a single slab drilled with holes for passages.<ref name="ref_edc5" />
 +
* Following up on the math-y stuff from my last post, I'm going to be taking a look at another concept that pops up in ML: manifolds.<ref name="ref_52d5">[http://bjlkeng.github.io/posts/manifolds/ Manifolds: A Gentle Introduction]</ref>
 +
* For example, all "cat images" might lie on a lower-dimensional manifold compared to say their original 256x256x3 image dimensions.<ref name="ref_52d5" />
 +
* Okay, that's all well and good, but that still doesn't answer the question: what is a manifold?<ref name="ref_52d5" />
 +
* A manifold is a topological space that "locally" resembles Euclidean space.<ref name="ref_52d5" />
 +
* The carburetor or the fuel injectors spray fuel droplets into the air in the manifold.<ref name="ref_0b5f">[https://en.wikipedia.org/wiki/Inlet_manifold Inlet manifold]</ref>
 +
* Comparison of a stock intake manifold for a Volkswagen 1.8T engine (top) to a custom-built one used in competition (bottom).<ref name="ref_0b5f" />
 +
* In the custom-built manifold, the runners to the intake ports on the cylinder head are much wider and more gently tapered.<ref name="ref_0b5f" />
 +
* This high-pressure air begins to equalize with lower-pressure air in the manifold.<ref name="ref_0b5f" />
 +
* To make use of the idea of a manifold a transition from the local to the global point of view is usually made.<ref name="ref_be20">[https://encyclopediaofmath.org/wiki/Manifold Encyclopedia of Mathematics]</ref>
 +
* For a disconnected manifold the components are usually taken to be of the same dimension.<ref name="ref_be20" />
 +
* A connected manifold without boundary is called open if it is non-compact, and closed if it is compact.<ref name="ref_be20" />
 +
* The global specification of a manifold is accomplished by an atlas: A set of charts covering the manifold.<ref name="ref_be20" />
 +
* The car's infotainment computer directs vehicle controllers that talk to valves that move the air through a manifold.<ref name="ref_781c">[https://www.merriam-webster.com/dictionary/manifold Definition of Manifold by Merriam-Webster]</ref>
 +
* As a result, the company had to shut down one manifold, which effectively branches into several lines carrying propellant to four thrusters.<ref name="ref_781c" />
 +
* One of the goals of topology is to find ways of distinguishing manifolds.<ref name="ref_08b6">[https://mathworld.wolfram.com/Manifold.html Manifold -- from Wolfram MathWorld]</ref>
 +
* For instance, a circle is topologically the same as any closed loop, no matter how different these two manifolds may appear.<ref name="ref_08b6" />
 +
* As a topological space, a manifold can be compact or noncompact, and connected or disconnected.<ref name="ref_08b6" />
 +
* Commonly, the unqualified term "manifold"is used to mean "manifold with boundary." This is the usage followed in this work.<ref name="ref_08b6" />
 +
* Here we will focus on the general notion of a manifold.<ref name="ref_4223">[https://ncatlab.org/nlab/show/manifold manifold in nLab]</ref>
 +
* At best, we can only talk about isomorphisms of manifolds.<ref name="ref_4223" />
 +
* An atlas is not considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure.<ref name="ref_4223" />
 +
* Morphisms of manifolds are here called smooth maps, and isomorphisms are called diffeomorphisms.<ref name="ref_4223" />
 +
* This step aims to approximate the manifolds of the datasets.<ref name="ref_3146">[https://bmcgenomics.biomedcentral.com/articles/10.1186/s12864-019-6329-2 ManiNetCluster: a novel manifold learning approach to reveal the functional links between gene networks]</ref>
 +
* Then, we cluster those networks simultaneously based on the distances in the common manifold.<ref name="ref_3146" />
 +
* I claim that a super useful step in answering this question is understanding what a manifold is.<ref name="ref_379f">[https://towardsdatascience.com/manifolds-in-data-science-a-brief-overview-2e9dde9437e5 Manifolds in Data Science — A Brief Overview]</ref>
 +
* Visualize examples of manifolds in various contexts.<ref name="ref_379f" />
 +
* To be a manifold, there’s one important rule that needs to be satisfied.<ref name="ref_379f" />
 +
* Suppose there is a small ant walking along a manifold in three dimensions.<ref name="ref_379f" />
 +
* The course will start by introducing the concept of a manifold (without recourse to an embedding into an ambient space).<ref name="ref_863f">[https://warwick.ac.uk/fac/sci/maths/undergrad/ughandbook/year3/ma3h5/ MA3H5 Manifolds]</ref>
 +
* Colour qualities form a two-dimensional manifold (cf.<ref name="ref_863f" />
 +
* In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.<ref name="ref_16c1">[https://en.wikipedia.org/wiki/Manifold Wikipedia]</ref>
 +
* Two-dimensional manifolds are also called surfaces.<ref name="ref_16c1" />
 +
* A Riemannian metric on a manifold allows distances and angles to be measured.<ref name="ref_16c1" />
 +
* A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point.<ref name="ref_16c1" />
 +
* A, B are the n by m PC matrices that span the task-specific manifolds A and B; the corresponding PC neural modes are their column vectors.<ref name="ref_c8a6">[https://www.nature.com/articles/s41467-018-06560-z Cortical population activity within a preserved neural manifold underlies multiple motor behaviors]</ref>
 +
* In dPCA, the rank m of the n by n matrix A is chosen as the desired dimensionality of the manifold.<ref name="ref_c8a6" />
 +
* As before, the chosen manifold dimensionality was m = 12, although the results held for m = 8, 15 (see Supplementary Fig.<ref name="ref_c8a6" />
 +
* Cognate with Middle High German manecvalt (“manifold”), Icelandic margfaldr (“multiple”).<ref name="ref_5f59">[https://en.wiktionary.org/wiki/manifold Wiktionary]</ref>
 +
* To make manifold; multiply.<ref name="ref_5f59" />
 +
* Direct mounted 2 valve manifold delivered with 2 bolts and one PTFE gasket.<ref name="ref_5edc">[https://www.fujielectric.fr/en/product/manifolds Manifold for pressure transmitter]</ref>
 +
* with pages giving succinct and precise of important concepts in the theory of manifolds.<ref name="ref_f26f">[http://www.map.mpim-bonn.mpg.de/ Manifold Atlas]</ref>
 +
* The term manifold is derived from Riemann's original German term, Mannigfaltigkeit.<ref name="ref_0857">[https://www.sciencedirect.com/topics/mathematics/differentiable-manifold Differentiable Manifold - an overview]</ref>
 +
* Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is formalised today as the concept of manifold.<ref name="ref_0857" />
 +
* A manifold, also a differentiable manifold, is defined as a topological space that is locally equivalent to the Euclidean space.<ref name="ref_0857" />
 +
* This amounts to say that each point of the manifold belongs to an open set which is homeomorphic to an open set of the Euclidean space.<ref name="ref_0857" />
 +
* As many of the results in the paper come from this embedding, it is important to actually note what the structure of this manifold is.<ref name="ref_8cb1">[https://elifesciences.org/articles/46409 The manifold structure of limb coordination in walking Drosophila]</ref>
 +
* Moreover, the density within the manifold is not shown in any of the plots as well.<ref name="ref_8cb1" />
 +
* Using this projection, we visualized the density of points within the manifold.<ref name="ref_8cb1" />
 +
* By construction, the high-dimensional data manifold produced by the model is continuous.<ref name="ref_8cb1" />
 +
===소스===
 +
<references />

2020년 12월 16일 (수) 11:00 기준 최신판

메모

  • homology manifolds
  • topological/smooth/PL manifolds


리뷰, 에세이, 강의노트


관련도서

  • James, I. M., ed. 1999. History of Topology. Amsterdam: North-Holland. http://www.ams.org/mathscinet-getitem?mr=1674906.
    • Scholz, Erhard. 1999. “The Concept of Manifold, 1850–1950.” In History of Topology, 25–64. Amsterdam: North-Holland.

노트

  • Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance.[1]
  • In the same vein, the Japanese word "多様体" (tayōtai) also encompasses both manifold and variety.[1]
  • The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness".[1]
  • Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold.[1]
  • The way these connect to one another dictates the control options of a manifold.[2]
  • A Drilled manifold, on the other hand, is made with a single slab drilled with holes for passages.[2]
  • Following up on the math-y stuff from my last post, I'm going to be taking a look at another concept that pops up in ML: manifolds.[3]
  • For example, all "cat images" might lie on a lower-dimensional manifold compared to say their original 256x256x3 image dimensions.[3]
  • Okay, that's all well and good, but that still doesn't answer the question: what is a manifold?[3]
  • A manifold is a topological space that "locally" resembles Euclidean space.[3]
  • The carburetor or the fuel injectors spray fuel droplets into the air in the manifold.[4]
  • Comparison of a stock intake manifold for a Volkswagen 1.8T engine (top) to a custom-built one used in competition (bottom).[4]
  • In the custom-built manifold, the runners to the intake ports on the cylinder head are much wider and more gently tapered.[4]
  • This high-pressure air begins to equalize with lower-pressure air in the manifold.[4]
  • To make use of the idea of a manifold a transition from the local to the global point of view is usually made.[5]
  • For a disconnected manifold the components are usually taken to be of the same dimension.[5]
  • A connected manifold without boundary is called open if it is non-compact, and closed if it is compact.[5]
  • The global specification of a manifold is accomplished by an atlas: A set of charts covering the manifold.[5]
  • The car's infotainment computer directs vehicle controllers that talk to valves that move the air through a manifold.[6]
  • As a result, the company had to shut down one manifold, which effectively branches into several lines carrying propellant to four thrusters.[6]
  • One of the goals of topology is to find ways of distinguishing manifolds.[7]
  • For instance, a circle is topologically the same as any closed loop, no matter how different these two manifolds may appear.[7]
  • As a topological space, a manifold can be compact or noncompact, and connected or disconnected.[7]
  • Commonly, the unqualified term "manifold"is used to mean "manifold with boundary." This is the usage followed in this work.[7]
  • Here we will focus on the general notion of a manifold.[8]
  • At best, we can only talk about isomorphisms of manifolds.[8]
  • An atlas is not considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure.[8]
  • Morphisms of manifolds are here called smooth maps, and isomorphisms are called diffeomorphisms.[8]
  • This step aims to approximate the manifolds of the datasets.[9]
  • Then, we cluster those networks simultaneously based on the distances in the common manifold.[9]
  • I claim that a super useful step in answering this question is understanding what a manifold is.[10]
  • Visualize examples of manifolds in various contexts.[10]
  • To be a manifold, there’s one important rule that needs to be satisfied.[10]
  • Suppose there is a small ant walking along a manifold in three dimensions.[10]
  • The course will start by introducing the concept of a manifold (without recourse to an embedding into an ambient space).[11]
  • Colour qualities form a two-dimensional manifold (cf.[11]
  • In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.[12]
  • Two-dimensional manifolds are also called surfaces.[12]
  • A Riemannian metric on a manifold allows distances and angles to be measured.[12]
  • A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point.[12]
  • A, B are the n by m PC matrices that span the task-specific manifolds A and B; the corresponding PC neural modes are their column vectors.[13]
  • In dPCA, the rank m of the n by n matrix A is chosen as the desired dimensionality of the manifold.[13]
  • As before, the chosen manifold dimensionality was m = 12, although the results held for m = 8, 15 (see Supplementary Fig.[13]
  • Cognate with Middle High German manecvalt (“manifold”), Icelandic margfaldr (“multiple”).[14]
  • To make manifold; multiply.[14]
  • Direct mounted 2 valve manifold delivered with 2 bolts and one PTFE gasket.[15]
  • with pages giving succinct and precise of important concepts in the theory of manifolds.[16]
  • The term manifold is derived from Riemann's original German term, Mannigfaltigkeit.[17]
  • Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is formalised today as the concept of manifold.[17]
  • A manifold, also a differentiable manifold, is defined as a topological space that is locally equivalent to the Euclidean space.[17]
  • This amounts to say that each point of the manifold belongs to an open set which is homeomorphic to an open set of the Euclidean space.[17]
  • As many of the results in the paper come from this embedding, it is important to actually note what the structure of this manifold is.[18]
  • Moreover, the density within the manifold is not shown in any of the plots as well.[18]
  • Using this projection, we visualized the density of points within the manifold.[18]
  • By construction, the high-dimensional data manifold produced by the model is continuous.[18]

소스

  1. ↑ 1.0 1.1 1.2 1.3 History of manifolds and varieties
  2. ↑ 2.0 2.1 The Benefits and Uses of Manifold Blocks
  3. ↑ 3.0 3.1 3.2 3.3 Manifolds: A Gentle Introduction
  4. ↑ 4.0 4.1 4.2 4.3 Inlet manifold
  5. ↑ 5.0 5.1 5.2 5.3 Encyclopedia of Mathematics
  6. ↑ 6.0 6.1 Definition of Manifold by Merriam-Webster
  7. ↑ 7.0 7.1 7.2 7.3 Manifold -- from Wolfram MathWorld
  8. ↑ 8.0 8.1 8.2 8.3 manifold in nLab
  9. ↑ 9.0 9.1 ManiNetCluster: a novel manifold learning approach to reveal the functional links between gene networks
  10. ↑ 10.0 10.1 10.2 10.3 Manifolds in Data Science — A Brief Overview
  11. ↑ 11.0 11.1 MA3H5 Manifolds
  12. ↑ 12.0 12.1 12.2 12.3 Wikipedia
  13. ↑ 13.0 13.1 13.2 Cortical population activity within a preserved neural manifold underlies multiple motor behaviors
  14. ↑ 14.0 14.1 Wiktionary
  15. ↑ Manifold for pressure transmitter
  16. ↑ Manifold Atlas
  17. ↑ 17.0 17.1 17.2 17.3 Differentiable Manifold - an overview
  18. ↑ 18.0 18.1 18.2 18.3 The manifold structure of limb coordination in walking Drosophila
원본 주소 "https://wiki.mathnt.net/index.php?title=다양체&oldid=45946"