Related papers: How to construct generalized van der Corput sequen…
In this paper we associate to any $LS$-sequence of partitions ${\rho_{L,S}^n}$ the corresponding $LS$-sequence of points ${\xi_{L,S}^n}$ obtained reordering the points of each partition with an explicit algorithm. The procedure begins with…
The main purpose of this master thesis is to study the $LS$-sequences of points introduced by Carbone in \cite{Carbone} and find two generalizations of them to the unit square. Here we also present a new algorithm proposed by the same…
Generalized L\"uroth series generalize $b$-adic representations as well as L\"uroth series. Almost all real numbers are normal, but it is not easy to construct one. In this paper, a new construction of normal numbers with respect to…
Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…
Standard sweep algorithms require an order of discrete points in Euclidean space, and rely on the property that, at a given point, all points in the halfspace below come earlier in this order. We are motivated by the problem of…
We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for…
In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van…
Champernowne famously proved that the number $0.(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)...$ formed by concatenating all the integers one after another is normal base 10. We give a generalization of Champernowne's construction to various…
In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For…
The $LS$-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone, who also proved that under an appropriate choice of the parameters $L$ and $S$, such sequences are low-discrepancy. The…
The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms-including Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as well as…
A new global basis of B-splines is defined in the space of generalized quadratic splines (GQS) generated by Merrien subdivision algorithm. Then, refinement equations for these B-splines and the associated corner-cutting algorithm are given.…
A new statistical technique for constructing linear latent structure (LLS) models from available data, supported by well established theoretical results and an efficient algorithm, is presented. The method reduces the problem of estimating…
Similarly to $\beta$-adic van der Corput sequences, abstract van der Corput sequences can be defined for abstract numeration systems. Under some assumptions, these sequences are low discrepancy sequences. The discrepancy function is…
This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any n-tuple occurs at most once in a period in either direction. As has been previously described, such…
In this work, we generalize the integer enumeration basis. We also construct bijections between the elements of special sets and the elements of some groups, and treat the special case of the hyperoctohedral groups. Then, we find a code…
This work discusses an approach to solving geometric construction problems in which the given figure is included in a set ordered by construction steps. The flow of information is carried through the chain, allowing the original problem to…
A coreset (or core-set) of an input set is its small summation, such that solving a problem on the coreset as its input, provably yields the same result as solving the same problem on the original (full) set, for a given family of problems…
We introduce a class of rooted graphs which allows one to encode various kinds of classical or quantum circuits. We then follow a set-theoretic approach to define rewrite systems over the considered graphs and propose a new complete…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…