# 다양체

(미분다양체에서 넘어옴)

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## 메모

- homology manifolds
- topological/smooth/PL manifolds

## 리뷰, 에세이, 강의노트

- Quinn, History of manifolds

## 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

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}History of manifolds and varieties - ↑
^{2.0}^{2.1}The Benefits and Uses of Manifold Blocks - ↑
^{3.0}^{3.1}^{3.2}^{3.3}Manifolds: A Gentle Introduction - ↑
^{4.0}^{4.1}^{4.2}^{4.3}Inlet manifold - ↑
^{5.0}^{5.1}^{5.2}^{5.3}Encyclopedia of Mathematics - ↑
^{6.0}^{6.1}Definition of Manifold by Merriam-Webster - ↑
^{7.0}^{7.1}^{7.2}^{7.3}Manifold -- from Wolfram MathWorld - ↑
^{8.0}^{8.1}^{8.2}^{8.3}manifold in nLab - ↑
^{9.0}^{9.1}ManiNetCluster: a novel manifold learning approach to reveal the functional links between gene networks - ↑
^{10.0}^{10.1}^{10.2}^{10.3}Manifolds in Data Science — A Brief Overview - ↑
^{11.0}^{11.1}MA3H5 Manifolds - ↑
^{12.0}^{12.1}^{12.2}^{12.3}Wikipedia - ↑
^{13.0}^{13.1}^{13.2}Cortical population activity within a preserved neural manifold underlies multiple motor behaviors - ↑
^{14.0}^{14.1}Wiktionary - ↑ Manifold for pressure transmitter
- ↑ Manifold Atlas
- ↑
^{17.0}^{17.1}^{17.2}^{17.3}Differentiable Manifold - an overview - ↑
^{18.0}^{18.1}^{18.2}^{18.3}The manifold structure of limb coordination in walking Drosophila