Let $${\bf A}$$ be an $$n\times m$$ matrix over $$\mathbf{GF}_2$$ where each column consists of $$k$$ ones, and let $$M$$ be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant $$C_M$$ such that $$m\geq C_M n^2$$ implies that the binary matroid induced by {\bf A} contains $$M$$ as a minor. We prove that if the columns of $${\bf A}={\bf A}_{n,m,k}$$ are chosen \emph{randomly}, then there are constants $$k_M, L_M$$ such that $$k\geq k_M$$ and $$m\geq L_M n$$ implies that $${\bf A}$$ contains $$M$$ as a minor w.h.p.
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A simple yet general model of binary diffusion coefficients emerged from a comprehensive assessment of 18 binary systems
- Award ID(s):
- 1904245
- PAR ID:
- 10298712
- Date Published:
- Journal Name:
- Acta Materialia
- Volume:
- 215
- Issue:
- C
- ISSN:
- 1359-6454
- Page Range / eLocation ID:
- 117077
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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