Computing Node Polynomials for Plane Curves

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

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

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

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

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

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

Beyond the Meta: Leveraging Game Design Parameters for Patch-Agnostic Esport Analytics

Esport games comprise a sizeable fraction of the global games market, and is the fastest growing segment in games. This has given rise to the domain of esports analytics, which uses telemetry data from games to inform players, coaches, broadcasters and other stakeholders. Compared to traditional sports, esport titles change rapidly, in terms of mechanics as well as rules. Due to these frequent changes to the parameters of the game, esport analytics models can have a short life-spam, a problem which is largely ignored within the literature. This paper extracts information from game design (i.e. patch notes) and utilises clustering techniques to propose a new form of character representation. As a case study, a neural network model is trained to predict the number of kills in a Dota 2 match utilising this novel character representation technique. The performance of this model is then evaluated against two distinct baselines, including conventional techniques. Not only did the model significantly outperform the baselines in terms of accuracy (85% AUC), but the model also maintains the accuracy in two newer iterations of the game that introduced one new character and a brand new character type. These changes introduced to the design of the game would typically break conventional techniques that are commonly used within the literature. Therefore, the proposed methodology for representing characters can increase the life-spam of machine learning models as well as contribute to a higher performance when compared to traditional techniques typically employed within the literature

Tropical convexity via cellular resolutions

Abstract The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be used to compute tropical convex hulls. Tropical cyclic polytopes are also presented