188 research outputs found
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
Relative Node Polynomials for Plane Curves
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
-Floor Diagrams computing Refined Severi Degrees for Plane Curves
The Severi degree is the degree of the Severi variety parametrizing plane curves of degree with nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable , which are conjecturally equal, for large . At , 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 -analog count of Brugallé and Mikhalkin's floor diagrams. Our description implies that, for fixed , the refined Severi degrees are polynomials in and , for large . As a consequence, we show that, for and all , both refinements of Göttsche and Shende agree and equal our -count of floor diagrams
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
- …