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For an $$r$$-uniform hypergraph $$H$$, let $$\nu^{(m)}(H)$$ denote the maximum size of a set $$M$$ of edges in $$H$$ such that every two edges in $$M$$ intersect in less than $$m$$ vertices, and let $$\tau^{(m)}(H)$$ denote the minimum size of a collection $$C$$ of $$m$$-sets of vertices such that every edge in $$H$$ contains an element of $$C$$. The fractional analogues of these parameters are denoted by $$\nu^{*(m)}(H)$$ and $$\tau^{*(m)}(H)$$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(r-1)}(H)/\nu^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$$. In this paper we prove bounds on the ratio between the parameters $$\tau^{(m)}$$ and $$\nu^{(m)}$$, and their fractional analogues. Our main result is that, for every $$r$$-uniform hypergraph~$$H$$,\[ \tau^{*(r-1)}(H)/\nu^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\\frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\\end{cases} \]This improves the known bound of $$r-1$.We also prove that, for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(m)}(H)/\nu^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$$, where the Turán number $$\operatorname{ex}_r(n, k)$$ is the maximum number of edges in an $$r$$-uniform hypergraph on $$n$$ vertices that does not contain a copy of the complete $$r$$-uniform hypergraph on $$k$$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$.
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A Synthetic Nervous System for on and Off Motion Detection Inspired by the Drosophila melanogaster Optic Lobe
Nourse, W. R., Szczecinski, N. S., & Quinn, R. D. (2023, July). A Synthetic Nervous System for on and Off Motion Detection Inspired by the Drosophila melanogaster Optic Lobe. In Conference on Biomimetic and Biohybrid Systems (pp. 364-380). Cham: Springer Nature Switzerland.
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- PAR ID:
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- Cham: Springer Nature Switzerland.
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- National Science Foundation
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