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1.
The true catenary
Tjaša Hrovatič, 2013, diplomsko delo

Opis: In this thesis we introduce the problem of the ideal homogeneous hanging cable called the catenary. We observe the behaviour of the shape of the curve. Firstly, we solve the problem of the classical catenary on a flfat Earth, where the gravitational fifield is constant and perpendicular to the ground. Secondly, we focus on the true symmetric catenary in the central gravitational fifield, which comes from the -1/r potential. In both cases we use the method of calculus of variations for isoperimetric problems and in particular the Euler-Lagrange difffferential equation. Lastly, we explain the problem of the asymmetric case.
Ključne besede: Catenary, calculus of variations, Euler-Lagrange equation, curvature, potential energy, difffferential equation.
Objavljeno: 25.09.2013; Ogledov: 734; Prenosov: 42
.pdf Polno besedilo (571,58 KB)

2.
The number of moves of the largest disc in shortest paths on Hanoi graphs
Simon Aumann, Katharina Götz, Andreas Hinz, Ciril Petr, 2014, izvirni znanstveni članek

Opis: In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs ▫$H_p^n$▫ in a more general setting. Here ▫$p$▫ stands for the number of pegs and ▫$n$▫ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length ▫$p-1$▫ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all ▫$n \geqslant p(p-2)$▫, of optimal paths where ▫$p-1$▫ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.
Ključne besede: graph theory, Tower of Hanoi, Hanoi graphs, shortest paths, symmetries, breadth-first search
Objavljeno: 14.08.2017; Ogledov: 81; Prenosov: 2
.pdf Polno besedilo (376,70 KB)

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