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UBC Theses and Dissertations
Some characteristics of the second betti number of random two dimensional simplicial complexes Tan, Kang
Abstract
In this thesis, through generating random two-dimensional simplicial complexes, we studied the event (b₂=0) for some specific probabilities. We found when the probability of event (b₂=0) takes on certain specific values, the pair (n₀,n₂) lies on certain lines. However, this research is limited by our sample space ( i.e. for P(b₂=0) ≈ 10%, 10 ≤ n₀ ≤ 80; for P(b₂=0) ≈ 50%, 12 ≤ n₀ ≤ 100; for P(b₂=0) ≈ 90%, 12 ≤ n₀ ≤ 145). The " linear behavior" may not hold asymptotically. In the same time, we endeavor to find the number of tetrahedra and 6-triangles in the simplicial complexes. When the event (b₂=0) occurs in our specific probabilities, it seems the second Betti number should come from tetrahedra and 6-triangles with high probability. However, the expectation of the number of tetrahedra and 6-triangles goes to zero, when no goes to infinity and there exists linear relationships between the pair (n₀,n₂). This evidence may also support that the " linear behavior" may not hold asymptotically. If n₂ and n₀ vary linearly with n₀ going to infinity, then the probability that n₀(Κ) — n₀ is extremely small in model MB for reasons are similar to the Coupon Collector's problem. Hence, the probability that we cannot find element in S(n₀,n₂) is large, which indicates that the model may have problems.
Item Metadata
Title |
Some characteristics of the second betti number of random two dimensional simplicial complexes
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1996
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Description |
In this thesis, through generating random two-dimensional simplicial complexes,
we studied the event (b₂=0) for some specific probabilities. We found when the
probability of event (b₂=0) takes on certain specific values, the pair (n₀,n₂) lies on
certain lines. However, this research is limited by our sample space ( i.e. for
P(b₂=0) ≈ 10%, 10 ≤ n₀ ≤ 80; for P(b₂=0) ≈ 50%, 12 ≤ n₀ ≤ 100; for P(b₂=0) ≈ 90%,
12 ≤ n₀ ≤ 145). The " linear behavior" may not hold asymptotically. In the same time,
we endeavor to find the number of tetrahedra and 6-triangles in the simplicial
complexes. When the event (b₂=0) occurs in our specific probabilities, it seems the
second Betti number should come from tetrahedra and 6-triangles with high probability.
However, the expectation of the number of tetrahedra and 6-triangles goes to zero,
when no goes to infinity and there exists linear relationships between the pair (n₀,n₂).
This evidence may also support that the " linear behavior" may not hold asymptotically.
If n₂ and n₀ vary linearly with n₀ going to infinity, then the probability that n₀(Κ) — n₀ is
extremely small in model MB for reasons are similar to the Coupon Collector's
problem. Hence, the probability that we cannot find element in S(n₀,n₂) is large, which
indicates that the model may have problems.
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Extent |
1502253 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2009-02-11
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0079898
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
1996-05
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.