| 1. |
- Goranko, Valentin, 1959-, et al.
(författare)
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STRUCTURAL THEORY OF TREES I. BRANCHING AND CONDENSATIONS OF TREES
- 2023
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Ingår i: Contributions to Discrete Mathematics. - 1715-0868. ; 18:2, s. 188-209
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Tidskriftsartikel (refereegranskat)abstract
- Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of branching in trees and of condensed trees, which are trees in which every node is a branching node. We then introduce and investigate two different constructions of tree condensations-one shrinking, and the other expanding, the tree to a condensed tree.
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| 2. |
- Kellerman, Ruaan, et al.
(författare)
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Structural theory of trees II. Completeness and completions of tre
- 2023
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Ingår i: Contributions to Discrete Mathematics. - 1715-0868. ; 18:2, s. 210-233
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Tidskriftsartikel (refereegranskat)abstract
- Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of tree completions that extend any tree to a minimal one satisfying the respective completeness property.
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| 3. |
- Conradie, Willem, et al.
(författare)
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Logic and Discrete Mathematics : a Concise Introduction
- 2015
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Bok (refereegranskat)abstract
- This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual.
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| 4. |
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| 5. |
- Fernández-Duque, David, et al.
(författare)
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Secure aggregation of distributed information : How a team of agents can safely share secrets in front of a spy
- 2016
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Ingår i: Discrete Applied Mathematics. - : Elsevier BV. - 0166-218X .- 1872-6771. ; 198, s. 118-135
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Tidskriftsartikel (refereegranskat)abstract
- We consider the generic problem of Secure Aggregation of Distributed Information (SADI), where several agents acting as a team have information distributed amongst them, modelled by means of a publicly known deck of cards distributed amongst the agents, so that each of them knows only her cards. The agents have to exchange and aggregate the information about how the cards are distributed amongst them by means of public announcements over insecure communication channels, intercepted by an adversary “eavesdropper”, in such a way that the adversary does not learn who holds any of the cards. We present a combinatorial construction of protocols that provides a direct solution of a class of SADI problems and develop a technique of iterated reduction of SADI problems to smaller ones which are eventually solvable directly. We show that our methods provide a solution to a large class of SADI problems, including all SADI problems with sufficiently large size and sufficiently balanced card distributions.
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