3,593 research outputs found
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Certified Context-Free Parsing: A formalisation of Valiant's Algorithm in Agda
Valiant (1975) has developed an algorithm for recognition of context free
languages. As of today, it remains the algorithm with the best asymptotic
complexity for this purpose. In this paper, we present an algebraic
specification, implementation, and proof of correctness of a generalisation of
Valiant's algorithm. The generalisation can be used for recognition, parsing or
generic calculation of the transitive closure of upper triangular matrices. The
proof is certified by the Agda proof assistant. The certification is
representative of state-of-the-art methods for specification and proofs in
proof assistants based on type-theory. As such, this paper can be read as a
tutorial for the Agda system
Comparing Machine Learning Algorithms by Union-Free Generic Depth
We propose a framework for descriptively analyzing sets of partial orders
based on the concept of depth functions. Despite intensive studies in linear
and metric spaces, there is very little discussion on depth functions for
non-standard data types such as partial orders. We introduce an adaptation of
the well-known simplicial depth to the set of all partial orders, the
union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a
comparison of machine learning algorithms based on multidimensional performance
measures. Concretely, we provide two examples of classifier comparisons on
samples of standard benchmark data sets. Our results demonstrate promisingly
the wide variety of different analysis approaches based on ufg methods.
Furthermore, the examples outline that our approach differs substantially from
existing benchmarking approaches, and thus adds a new perspective to the vivid
debate on classifier comparison.Comment: arXiv admin note: substantial text overlap with arXiv:2304.0987
- …