# 다양체

(미분다양체에서 넘어옴)
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## 메모

• homology manifolds
• topological/smooth/PL 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]