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Theory of incompatible substructure problem

2014-11-27T22:31:07Z

Awinslow: Added some open problems.

== Schedule ==
* 19th December, 2014 : Submit final draft of incompatible substructure paper to Marco
* 30th November, 2014 : Finish results, create plots

== Todo ==

== Parameters ==
* Components
** Radius of each component: 25 mm
** Thickness of each component: '''8 mm'''
** Radius of magnet: 1.5 mm
** Strength of magnet: N48
** Polyamide

* Container
** Radius of container: 125 mm
** Depth of container: 9 mm
** Material: Acrylic

* Shaker
** Mode: Orbital shaking
** Speed: 300 rpm
** Duration for shaking: until stable structures are formed

== Variable(s) ==
* Number of components used in each experiment

== Experiments ==
* Simulated experiments
** Probability of the formation of 1, 2, 3, 4 and 5 targets for increasing number of components
* Physical experiments
** Probability of the formation of 1, 2, 3, 4 and 5 targets for increasing number of components

=== Particulars of simulation experiments (10000 trials each) ===
* Probability of the formation of 1 target structure in experiments with increasing number of components from 8 - 40
* Probability of the formation of 2 target structure in experiments with increasing number of components from 16 - 40
* Probability of the formation of 3 target structure in experiments with increasing number of components from 24 - 40
* Probability of the formation of 4 target structure in experiments with increasing number of components from 32 - 40
* Probability of the formation of 5 target structure in experiments with 40 components

=== Particulars of physical experiments (10 trials each) ===
* Photos of initial condition and final condition
* Assembly of one target structure
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of two target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of three target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of four target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of five target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state

* Results to show
** The simulated probability of the formation of 1, 2, 3, 4 and 5 target structures in self-assembly experiments
** Comparison of simulated probability vs. probability achieved in physical experiments
** Reasons for similarities and/or differences between simulated probability vs. probability of physical experiments

== Open problems ==

Using the model of Hosokawa 1995. The system starts as a set of n components. Then the following step is repeated until no longer possible: pick a random pair of structures with total size at most the size of a target structure. Combine them. The process stops when no such pair exists, i.e. when the two smallest structures have total size more than the size of a target structure.

Consider a system of n components that form target structures of size k. Let f(n, k, i) be the probability that i target substructures form.

* Probabilities of yields.
** f(n, k, 1) + f(n, k, 2) + ...: What is the probability that at least one target structure forms?
** f(n, k, floor(n/k)): What is the probability that the maximum number of target substructures form?
** f(n, k, 1) * 1 + f(n, k, 2) * 2 + ...: What is the average number of target substructures that form?
** General closed form for f(n, k, i): what is the probability that i substructures form for some fixed i?

* Unexpected behavior.
** Prove that f(n, k, m) is non-increasing as a function of n.

== Meeting points ==
* In the case of the experiment where 24 components are used, if 2 target structures are formed (16 components consumed), then the third target structure will always form. This means that if the experiment is continued till the system reaches a steady state, then in the experiment with 24 components, there will never be a case where 2 target structures are formed.
** Doesn't this skew the probability graph? Also, how would this be reflected in the analysis?
** Obviously, this applies to the cases of 2, 4, 5 target structures (16, 32, 40 components) as well.

Theory of incompatible substrucutre problem

2014-11-27T22:01:58Z

Awinslow: Awinslow moved page Theory of incompatible substrucutre problem to Theory of incompatible substructure problem: Typo in title.

#REDIRECT [[Theory of incompatible substructure problem]]

Theory of incompatible substructure problem

2014-11-27T22:01:58Z

Awinslow: Awinslow moved page Theory of incompatible substrucutre problem to Theory of incompatible substructure problem: Typo in title.

== Schedule ==
* 19th December, 2014 : Submit final draft of incompatible substructure paper to Marco
* 30th November, 2014 : Finish results, create plots

== Todo ==

== Parameters ==
* Components
** Radius of each component: 25 mm
** Thickness of each component: '''8 mm'''
** Radius of magnet: 1.5 mm
** Strength of magnet: N48
** Polyamide

* Container
** Radius of container: 125 mm
** Depth of container: 9 mm
** Material: Acrylic

* Shaker
** Mode: Orbital shaking
** Speed: 300 rpm
** Duration for shaking: until stable structures are formed

== Variable(s) ==
* Number of components used in each experiment

== Experiments ==
* Simulated experiments
** Probability of the formation of 1, 2, 3, 4 and 5 targets for increasing number of components
* Physical experiments
** Probability of the formation of 1, 2, 3, 4 and 5 targets for increasing number of components

=== Particulars of simulation experiments (10000 trials each) ===
* Probability of the formation of 1 target structure in experiments with increasing number of components from 8 - 40
* Probability of the formation of 2 target structure in experiments with increasing number of components from 16 - 40
* Probability of the formation of 3 target structure in experiments with increasing number of components from 24 - 40
* Probability of the formation of 4 target structure in experiments with increasing number of components from 32 - 40
* Probability of the formation of 5 target structure in experiments with 40 components

=== Particulars of physical experiments (10 trials each) ===
* Photos of initial condition and final condition
* Assembly of one target structure
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of two target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of three target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of four target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state
* Assembly of five target structures
** Data to be collected: number of targets and number of incompatible substructures, time for steady state

* Results to show
** The simulated probability of the formation of 1, 2, 3, 4 and 5 target structures in self-assembly experiments
** Comparison of simulated probability vs. probability achieved in physical experiments
** Reasons for similarities and/or differences between simulated probability vs. probability of physical experiments

== Meeting points ==
In the case of the experiment where 24 components are used, if 2 target structures are formed (16 components consumed), then the third target structure will always form. This means that if the experiment is continued till the system reaches a steady state, then in the experiemnt with 24 components, there will never be a case where 2 target structures are formed. '''Doesn't this skew the probability graph? Also, how would this be reflected in the analysis? ''' . Obviously, this applies to the cases of 2, 4 and 5 target structures as well.