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