31 research outputs found

    Cubic symmetric graphs of order a small number times a prime or a prime square

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    AbstractA graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K3,3 and the s-regular cyclic or elementary abelian coverings of the complete graph K4 for each s⩾1 are classified when the fibre-preserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regular graphs with girth 12 is found, in which the smallest one has order 2058. As an interesting application, a complete list of pairwise non-isomorphic s-regular cubic graphs of order 4p, 6p, 4p2 or 6p2 is given for each s⩾1 and each prime p

    A Victorian Age Proof of the Four Color Theorem

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    In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be rule out by re-constructing special 2-colored two paths decomposition in the form of a double-spiral chain of the maximal planar graph. In the second part of the paper we have given an algorithmic proof of the four color theorem which is based only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio

    Cubic symmetric graphs of order twice an odd prime-power

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    Contributions to Open Problems on Cable Driven Robots and Persistent Manifolds for the Synthesis of Mechanisms

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    Although many efforts are continuously devoted to the advancement of robotics, there are still many open and unresolved problems to be faced. This thesis, therefore, sets out to tackle some of them with the aim of scratching the surface and look a little further for new ideas or solutions. The topics covered are mainly two. The first part deals with the development and improvement of control techniques for cable-driven robots. The second focuses on the study of persistent manifolds seen as constituting aspects of theoretical kinematics. In detail, -Part I deals with cable-driven platforms. In it, both techniques for selecting cable tensions and the design of a robust controller are developed. The aim is, therefore, to enhance the two building blocks of the overall control scheme in order to improve the performance of these robots during the execution of tracking tasks. -- The first chapter introduces to open problems and recalls the main concepts necessary to understand the following chapters; -- the contribution of the second chapter consists of the introduction of the Analytic Centre. It allows the generation of continuous and differentiable tension profiles while taking into account non-linear phenomena such as friction in the computation of tensions to be applied; -- the third chapter, although still at a preliminary stage, introduces sensitivity for tension calculation methods, offering perspectives of considerable interest for tension control in the current scientific context; -- the fourth chapter proposes the design of an adaptive controller. It allows external disturbances and/or uncertainties in the model to be faced such that the task can be performed with as little error as possible. The controller architecture is the innovative peculiarity conferring autonomy to cable systems. Initially applied to counteract wind in aerial systems it is now also used for cable breakage scenarios; -- the conclusions, at first, draw together the results obtained. In addition, they emphasise the lack of the techniques introduced in order to outline possible future paths and topics that need further investigation. - Part II delves into theoretical kinematics. The discovery and classification of invariant screw systems shed light on numerous aspects of robot mobility and synthesis. Nevertheless, this generated the emergence of new ideas and questions that are still unresolved. Among them, one of the more notable concerns the identification and classification of 5-dimensional persistent manifolds. -- Similarly to the first part, the first chapter provides an overview of the problems addressed and the theoretical notions necessary to understand the subsequent contributions; -- the second chapter contributes by directly tackling the above-mentioned question by exploiting the properties of dual quaternions, the Study quadric and differential geometry. A library of 5-persistent varieties, so far missing in the literature, is presented along with theorems that complete and generalise previous ones in the literature; -- an original work, concerning line motions and synthesis of mechanisms that generate them, is reported in the third chapter as a spin-off of the studies on persistent manifolds; -- the conclusions wrap up the obtained results trying to highlight gaps and deficiencies to be dealt with in the future. Here, two small sections are dedicated to ongoing works regarding the persistence definition and the screw systems' invariants and subvariants
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