5,711 research outputs found

    Among graphs, groups, and latin squares

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    A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

    Graph Theory

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    Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

    Master index of volumes 161–170

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    A multi-model X-FEM strategy dedicated to frictional crack growth under cyclic fretting fatigue loadings.

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    International audienceA 2D X-FEM/LATIN numerical model (eXtended Finite Element Method/Large Time Increment method) is proposed in this paper for the analysis of fretting fatigue problems and the simulation of the crack propagation under such loadings. The half-analytical two-body contact analysis allows to capture accurately the pressure and the cyclic tractions exerted at the interface that induce non-proportional multi-axial loading. These distributions are then used as input data for determining critical location for crack initiation and crack inclination based on Dang Van's criterion. The frictional contact conditions of the fretting fatigue cracks have an important impact on the crack behaviour. In this respect, contact with friction between the crack faces is finely modeled within the X-FEM frame. The obtained results are compared and validated with a half-analytical reference model. The numerical simulations reveal the robustness and the efficiency of the proposed approach for a wide range of fretting loadings and friction coefficients values along crack faces. The crack growth directions are then predicted accurately based on the use of criteria adapted to multi-axial non-proportional fatigue. Four cases dealing with crack propagation are then presented. It is shown how the crack length, the tangential loading modify the crack path during the propagation process

    Glosarium Matematika

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    273 p.; 24 cm

    Glosarium Matematika

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