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EnglishIn this paper we investigate bounded additivity in Discrete Tomography. This notion has been previously introduced in [5], as a generalization of the original one in [11], which was given in terms of ridge functions. We exploit results from [6-8] to deal with bounded S non-additive sets of uniqueness, where S⊂Zn contains d coordinate directions {e_1,.., e_d}, |S|=d+1, and n≥d≥3. We prove that, when the union of two special subsets of {e_1,.., e_d} has cardinality k=n, then bounded S non-additive sets of uniqueness are confined in a grid A having a suitable fixed size in each coordinate direction ei, whereas, if k<n, the grid A can be arbitrarily large in each coordinate direction ei, where i>k. The subclass of pure bounded S non-additive sets plays a special role. We also compute explicitly the proportion of bounded S non-additive sets of uniqueness w.r.t. those additive, as well as w.r.t. the S-unique sets. This confirms a conjecture proposed by Fishburn et al. in [14] for the class of bounded sets.
| Autori: | |
| Titolo: | On bounded additivity in discrete tomography |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.tcs.2015.11.022 |
| Data di pubblicazione: | 2016 |
| Abstract: | In this paper we investigate bounded additivity in Discrete Tomography. This notion has been previously introduced in [5], as a generalization of the original one in [11], which was given in terms of ridge functions. We exploit results from [6-8] to deal with bounded S non-additive sets of uniqueness, where S⊂Zn contains d coordinate directions {e_1,.., e_d}, |S|=d+1, and n≥d≥3. We prove that, when the union of two special subsets of {e_1,.., e_d} has cardinality k=n, then bounded S non-additive sets of uniqueness are confined in a grid A having a suitable fixed size in each coordinate direction ei, whereas, if k<n, the grid A can be arbitrarily large in each coordinate direction ei, where i>k. The subclass of pure bounded S non-additive sets plays a special role. We also compute explicitly the proportion of bounded S non-additive sets of uniqueness w.r.t. those additive, as well as w.r.t. the S-unique sets. This confirms a conjecture proposed by Fishburn et al. in [14] for the class of bounded sets. |
| Lingua: | Inglese |
| Rivista: | |
| Citazione: | Brunetti, S. M., Dulio, P., Peri, C., On bounded additivity in discrete tomography, <<THEORETICAL COMPUTER SCIENCE>>, 2016; 2016/624 (Aprile): 89-100. [doi:10.1016/j.tcs.2015.11.022] [http://hdl.handle.net/10807/75662] |
| Appare nelle tipologie: | Articolo in rivista, Nota a sentenza |
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