Research Publication (math): Topology and Anomaly/Damage Detection


Just some simple notes about a powerful application of basic topology.




 Topology Publication



Restricted homeomorphism


Enclosing k-dimensional area/length/volume stays the same when the homeomorphism is applied


You have to define enclosure carefully.






What are restricted homeomorphisms good for in real life?


I can morph a shape without altering its surface area.


I can modify the curvature of a shape while leaving the rest of it intact.


What about in 4-D space time?


What is surface area in space time?




It’s any homeomorphism that preserves the sum of surface area of the morphed object over time.


You could, with calculus, devise a restricted homeomorphism that preserves inner area within the enclosure instead of surface area…or you could preserve both.


What about a “normal movement over time” homeomorphism, for something like a car or a stress ball?


A “mostly normal and passive movement homeomorphism “—for checking the status of an equipment item or a person.


To account for tiny changes, you could have a notion of distance from such a modified homeomorphism.


The idea is, consider the maximum distance from the homeomorphism over time.  Allow arbitrarily small time steps, as though you are working with limits.  I.e., the distance from the homeomorphism should be "correct" and within bounds for 1 unit, 1/2 unit, 1/4 unit, 1/8 unit, ... , ad infinitum.





I think topology is best for checking physical systems for anomalies.  The “acceptable“ changes to the physical system are part of the homeomorphism-like concept definition, and the bad/anomalous ones are outside the definition of the version of homeomorphism.


You could do automated review of a large volume of video footage with this.






You might be able to use software to automatically scan 3D video footage of say your or an adversary’s aircraft or personnel, to see if they’ve been damaged.  You might even be able to use it in conjunction with physics related AI to strategize on how to destroy your adversary’s assets without letting yours get damaged.


That is a big key application of topology.


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