25 research outputs found

    A Hilton-Milner theorem for vector spaces

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    We show for k = 2 that if q = 3 and n = 2k + 1, or q = 2 and n = 2k + 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF(q) with nFÂżF F = 0 has size at most (formula). This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding q-Kneser graphs

    PERC floods following “Bernd”

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    The Bernd floods, the severity of which have been linked to climate change, came at a time when climate change was and continues to be at the center of national and international political debates. Not only did the event highlight the urgency to address the climate crisis by drastically reducing greenhouse gas emissions, it also raised the question about limits to and failures of DRM and climate change adaptation. As traditional approaches are demonstrably not enough, how can countries and communities adapt to the new realities of climate change? If more transformational approaches are needed, what could they look like? In this report, we provide both key insights and concrete recommendations drawn from this flood event. Preparing for the future requires that we learn the lessons; and, learn not just for those areas that were affected this time, but in particular areas with similarities to the areas most impacted by Bernd, areas that could suffer similar losses in a future flood. It is especially those areas that must take action now to get to a higher preparedness level. As we have seen in the devastated areas, planning for reconstruction and implementing a forward-looking approach at the same time is nearly impossible as the affected population wants to get back to normal. Hence, often, opportunities are missed to improve and build forward - which needs to change

    Irregular weighting of 1-designs

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    Assign positive integer weights to the edges of a hypergraph in such a way that summing up the weights of the edges through a point yields distinct integers for different points. In this note we give a lower bound for the maximal edgeweight in case the hypergraph is uniform and regular, i.e. it is a 1-design. If the hypergraph is the dual of a 2-(v,k,Âż) design then this bound specializes to . In particular for a projective plane this number is at least

    Sets with a large number of nuclei on a conic

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    For large q we characterize the (q + 1)-sets in PG(2, q), q odd, having more than 0.326q nuclei on a conic. In the process more information about the structure of (q + 1)-sets having a large number of nuclei on a conic is obtained

    Proof of a conjecture by Đoković on the PoincarĂ© series of the invariants of a binary form

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    Ðokovic (2006) [3] gave an algorithm for the computation of the Poincaré series of the algebra of invariants of a binary form, where the correctness proof for the algorithm depended on an unproven conjecture. Here we prove this conjecture. Keywords: Ðokovic conjecture; Dixmier conjecture; Partial fraction decompositio

    The number of directions determined by a function f on a finite field

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    We give a short proof of RĂ©dei's result on the number of directions determined by a function f on a finite field and improve this result considerably, confirming one of his conjectures

    Caps Embedded in Grassmannians

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    This paper is concerned with constructing caps embedded in line Grassmannians. In particular, we construct a cap of size q 3 + 2q 2 + 1 embedded in the Klein quadric of PG(5; q) for even q, and show that any cap maximally embedded in the Klein quadric which is larger than this one must have size equal to the theoretical upper bound, namely q 3 + 2q 2 + q + 2. It is not known if caps achieving this upper bound exist for even q ? 2. 1 Introduction In [7] Glynn showed that any full Singer line orbit in PG(3; q) corresponds to a cap of size q 3 +q 2 +q+1 embedded in the Klein quadric K of PG(5; q). Moreover, for odd q he observed that this is the largest possible cap embedded in K. In this paper we show that larger caps can be embedded in K for even q, and we explicitly construct several infinite families of caps maximally embedded in K. The problem of completing caps to maximum caps on K is also addressed. Finally, we extend Glynn's idea to higher dimensions, thereby constru..
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