| 1. |
- Haukkanen, Pentti, et al.
(författare)
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The Arithmetic Jacobian Matrix and Determinant
- 2017
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Ingår i: Journal of Integer Sequences. - : University of Waterloo. - 1530-7638. ; 20:9
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Tidskriftsartikel (refereegranskat)abstract
- Let a1,..., am be real numbers that can be expressed as a finite product of prime powers with rational exponents. Using arithmetic partial derivatives, we define the arithmetic Jacobian matrix Ja of the vector a = (a1,..., am) analogously to the Jacobian matrix Jf of a vector function f. We introduce the concept of multiplicative independence of {a1,..., am} and show that Ja plays in it a similar role as Jf does in functional independence. We also present a kind of arithmetic implicit function theorem and show that Ja applies to it somewhat analogously as Jf applies to the ordinary implicit function theorem.
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| 2. |
- Knorn, Steffi, et al.
(författare)
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Quantitative analysis of curricula coherence using directed graphs
- 2019
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Ingår i: 12th IFAC Symposium on Advances in Control Education ACE 2019. - : Elsevier. ; 52:9, s. 318-323
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Konferensbidrag (refereegranskat)abstract
- This paper investigates methods for quantitatively examining the connectivity and knowledge flow in a university program considering courses and concepts included in the program. The proposed method is expected to be useful to aid program design and inventory, and for communicating what concepts a course may rely on at a given point in the program. As a first step, we represent the university program as a directed graph with courses and concepts as nodes and connections between courses and concepts as directed edges. Then, we investigate the connectivity and the flow through the graph in order to gain insights into the structure of the program. We thus perform two investigations based on data collected from an engineering program at a Swedish university: a) how to represent (parts of) the university program as a graph (here called Directed Courses-Concepts Graph (DCCG)), and b) how to use graph theory tools to analyse the coherence and structure of the program.
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| 3. |
- Mattila, Mika, et al.
(författare)
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Maximal perpendicularity in certain Abelian groups
- 2017
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Ingår i: Acta Universitatis Sapientiae, Mathematica. - : Walter de Gruyter. - 2066-7752. ; 9:1, s. 235-247
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Tidskriftsartikel (refereegranskat)abstract
- We define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (P, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.
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| 4. |
- Murua, M., et al.
(författare)
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Solving the multi-objective Hamiltonian cycle problem using a Branch-and-Fix based algorithm
- 2022
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Ingår i: Journal of Computational Science. - : Elsevier. - 1877-7503 .- 1877-7511. ; 60
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Tidskriftsartikel (refereegranskat)abstract
- The Hamiltonian cycle problem consists of finding a cycle in a given graph that passes through every single vertex exactly once, or determining that this cannot be achieved. In this investigation, a graph is considered with an associated set of matrices. The entries of each of the matrix correspond to a different weight of an arc. A multi-objective Hamiltonian cycle problem is addressed here by computing a Pareto set of solutions that minimize the sum of the weights of the arcs for each objective. Our heuristic approach extends the Branch-and-Fix algorithm, an exact method that embeds the problem in a stochastic process. To measure the efficiency of the proposed algorithm, we compare it with a multi-objective genetic algorithm in graphs of a different number of vertices and density. The results show that the density of the graphs is critical when solving the problem. The multi-objective genetic algorithm performs better (quality of the Pareto sets) than the proposed approach in random graphs with high density; however, in these graphs it is easier to find Hamiltonian cycles, and they are closer to the multi-objective traveling salesman problem. The results reveal that, in a challenging benchmark of Hamiltonian graphs with low density, the proposed approach significantly outperforms the multi-objective genetic algorithm.
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| 5. |
- Tossavainen, Timo, et al.
(författare)
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We Can Differentiate Numbers, Too
- 2024
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Ingår i: The College Mathematics Journal. - : Taylor & Francis. - 0746-8342 .- 1931-1346. ; 55:2, s. 100-108
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Tidskriftsartikel (refereegranskat)abstract
- We survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach’s conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing supplementary material for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises.
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