https://mp.weixin.qq.com/s/zoMJpYQeAAt6GmjPDx863Q
Just modify only 4 roads, the above complicated city road congestion improved : by Topology Dual Graphs.
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Tag Archives: Topology
Topology Math applied in Physics & Quantum Computing
Maths -Topologie
【区别：代数拓扑 (Algebraic Topology) 微分拓扑 (Differential Topology ) 微分几何 ( Differential Geometry ) 代数几何 (Algebraic Geometry ) 交换代数 (Commutative Algebra ) 微分流形 (Differential Manifold )
【区别：代数拓扑 (Algebraic Topology) 微分拓扑 (Differential Topology ) 微分几何 ( Differential Geometry ) 代数几何 (Algebraic Grometry ) 交换代数 (Commutative Algebra ) 微分流形 (Differential Manifold ) ？】月如歌：并不能理解什么叫做楼主所说的配对。我简要谈下我对于上述所列名词的理解。…
http://www.zhihu.com/question/23848852/answer/26771912 （分享自知乎网）
Sheaves do not belong to algebraic geometry:
https://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html
Who cares about topology? (inscribed rectangle problem)
Excellent video for the curious minds! Who cares about Topology such as Torus (aka donut) or Mobius Strip ? They can be used to prove difficult math such as the unsolved problem “Inscribed square/rectangle inside any closed loop”.
To understand the Topology on Loops, please view the lecture here : Homotopy (同伦) and the Fundamental Group (群) of surface.
Topology Game: 双人脱困
拓扑: 两绳困绑, 只要一人翻手就能解脱。
Topology Homology Theory – A Primer
Is the Abstract Mathematics of Topology Applicable to the Real World?
1st speaker :
◇ History: Riemann discovered Topology on his papers left behind after death. He told friend Betti.
◇ Betti Number: number of
– ‘scissors’ cut to make a tree (in 2 dim),
– ‘drill’ cut to make a disc (in 3 dim).
2nd speaker:
◇ Evolution (bacteria, viruses) using Topology ‘Barcoding’ technique.
比喻:
病菌(viruses / bacteria)是白骨精多变 (mutate), Topology (拓扑)是孙悟空的火眼金睛, 妖怪不能遁形, 骗得了常人, 骗不过老孙, “三打白骨精”。
3rd speaker:
◇ Liquid Crystal: Homology
Analysis -> (Topology) -> Algebra
Mathematics is divided into 2 major branches:
1. Analysis (Continuity, Calculus)
2. Algebra (Set, Discrete numbers, Structure)
In between the two branches, Poincaré invented in 1900s the Topology (拓扑学) – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.
Topology specialised in
– ‘local knowledge’ = Point-Set Topology.
– ”global knowledge’ = Algebraic Topology.
Example:
The local data of consumer behavior uses ‘Point-Set Topology’; the global one is ‘BIG Data’ using Algebraic Topology.
Topology & DNA
Turn Sphere Inside Out
Watch this amazing video, mathematically you can turn a sphere inside out, but not a circle:
http://www.snotr.com/video/3107/How_to_turn_a_sphere_inside_out
Hairy Ball Theorem
‘Hairy Ball Theorem’ (Topology)
1. “You can’t comb the hairs on a coconut.”
2. “There exists a calm point on earth where there is no wind.”
Both 1 & 2 are ‘Hairy Ball Theorem’.
If we have a ball with hairs sticking out from each point on it, then impossible to comb the hairs flat with a continuous movement without leaving at least one hair sticking up vertically (typically, at the center of a swirl). Not applicable to doughnut.
=> Wind pattern:
Ball = Earth
Combed down hairs = wind
Length of hairs = wind’s force
=> there must be a place on Earth where there is no wind at all.
Topology
Topology (by Poincaré)
Moniker “Rubber-Sheet Geometry“, compared with Geometry’s ‘rigid objects‘.
[Greek]= τοΠοζ(Place) λΟγια(Study)
[Latin]= Analysis Situs (Situation)
1. Remove (invariants) of geometry:
- a. Euclidean (distance)
- b. Affine (//, ratio)
- c. Projective (cross-ratio)
2. Preserve ‘Neighbourhood’ (Nearness)
- define ‘Continuity’ (Analysis)
3. Elastic deformation (stretch, bend, twist)
- a line is no longer a line.
