188 research outputs found

    Computing Node Polynomials for Plane Curves

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    According to the G\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify G\"ottsche's conjecture on the optimal threshold up to delta <= 14. We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.Comment: 23 pages; to appear in Mathematical Research Letter

    Relative Node Polynomials for Plane Curves

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    We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and delta nodes is given by a polynomial in d, provided delta is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined "relative node polynomial" in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary delta, and use it to present explicit formulas for delta <= 6. We also give a threshold for polynomiality, and compute the first few leading terms for any delta.Comment: 27 pages, final version, to be published in Journal of Algebraic Combinatoric

    Pen and paper techniques for physical customisation of tabletop interfaces

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    Win Prediction in Esports: Mixed-Rank Match Prediction in Multi-player Online Battle Arena Games

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    Esports has emerged as a popular genre for players as well as spectators, supporting a global entertainment industry. Esports analytics has evolved to address the requirement for data-driven feedback, and is focused on cyber-athlete evaluation, strategy and prediction. Towards the latter, previous work has used match data from a variety of player ranks from hobbyist to professional players. However, professional players have been shown to behave differently than lower ranked players. Given the comparatively limited supply of professional data, a key question is thus whether mixed-rank match datasets can be used to create data-driven models which predict winners in professional matches and provide a simple in-game statistic for viewers and broadcasters. Here we show that, although there is a slightly reduced accuracy, mixed-rank datasets can be used to predict the outcome of professional matches, with suitably optimized configurations

    Relative Node Polynomials for Plane Curves

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    We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ

    qq-Floor Diagrams computing Refined Severi Degrees for Plane Curves

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    The Severi degree is the degree of the Severi variety parametrizing plane curves of degree dd with δ\delta nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable qq, which are conjecturally equal, for large dd. At q=1q=1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a combinatorial description of the refined Severi degrees, in terms of a qq-analog count of Brugallé and Mikhalkin's floor diagrams. Our description implies that, for fixed δ\delta, the refined Severi degrees are polynomials in dd and qq, for large dd. As a consequence, we show that, for δ4\delta \leq 4 and all dd, both refinements of Göttsche and Shende agree and equal our qq-count of floor diagrams

    DETERMINING THE FIRE RATING OF CONCRETE STRUCTURES, Case study of using a probabilistic approach and travelling fires

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    As part of a refurbishment the height of a building in London is to be increased resulting in a change of the fire rating of the existing level from R60 to R90 as per prescriptive guidance. To investigate whether the inherent fire resistance of the structure would be sufficient a state-of-the-art probabilistic approach was adopted, with the approach extended to consider 2D heat-transfer to concrete elements. After determining the required reliability of the structure based on an acceptable risk level, a Monte-Carlo assessment was conducted. This considered for the proposed internal layouts and determined the range of input parameters to be randomly varied in order to define the required range of design fires analysed. The assessment demonstrated that the inherent structural fire resistance would provide sufficient structural reliability for the new use of the building and that no additional fire protection was required to the concrete frame
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