9 research outputs found
Exact arithmetic on the Stern–Brocot tree
AbstractIn this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials
Enumerating the rationals from left to right
Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown
to be generated via almost identical second order recurrence relations. These
sequences have combinatorial, computational, and geometric applications, and
are useful for enumerating the rational numbers.Comment: 6 pages, 3 figure
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Special values of zeta functions and areas of triangles
In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli spaces (for example of Abelian varieties) and their universal spaces
On the degree distribution of Haros graphs
Haros graphs is a graph-theoretical representation of real numbers in the
unit interval. The degree distribution of the Haros graphs provides information
regarding the topological structure and the associated real number. This
article provides a comprehensive demonstration of a conjecture concerning the
analytical formulation of the degree distribution. Specifically, a theorem
outlines the relationship between Haros graphs, the corresponding continued
fraction of its associated real number, and the subsequent symbolic paths in
the Farey binary Tree. Moreover, an expression continuous and piece-wise linear
in subintervals defined by Farey fractions can be derived from an additional
conclusion for the degree distribution of Haros graphs
Worst case and average case cardinality of strictly acute stencils for two dimensional anisotropic fast marching
We study a one dimensional approximation-like problem arising in the discretization of a class of Partial Differential Equations, providing worst case and average case complexity results. The analysis is based on the Stern-Brocot tree of rationals, and on a non-Euclidean notion of angles. The presented results generalize and improve on earlier work