Rendiconti dell'Istituto di Matematica dell'Universita di Trieste
In this paper we will generalize to higher dimension the splitting procedure introduced by Kakuta... more In this paper we will generalize to higher dimension the splitting procedure introduced by Kakutani for [0, 1]. This method will provide a sequence of nodes belonging to [0, 1] d which is uniformly distributed. The advantage of this approach is that it is intrinsecally d-dimensional.
In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed s... more In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed sequences of partitions. Among them we found a large class having low discrepancy (which means of order 1 N). One of them is the Kakutani-Fibonacci sequence.
We define a countable family of sequences of points in the unit square: the LS-sequences of point... more We define a countable family of sequences of points in the unit square: the LS-sequences of pointsà la Halton. They reveal a very strange and interesting behaviour, as well as resonance phenomena, for which we have not found an explanation, so far. We conclude with three open problems.
The LS-sequences of points recently introduced by the author are a generalization of van der Corp... more The LS-sequences of points recently introduced by the author are a generalization of van der Corput sequences. They were constructed by reordering the points of the corresponding LS-sequences of partitions. Here we present another algorithm which coincides with the classical one for van der Corput sequences and is simpler to compute than the original construction. This algorithm is based on the representation of natural numbers in base L + S and gives the van der Corput sequence in base b if L = b and S = 0. In this construction, as well as in the van der Corput one, it is essential the inversion of digits of the representation in base L + S: in this paper we also give a nice geometrical explanation of this "magical" operation.
Rendiconti del Circolo Matematico di Palermo, 2011
In this paper we consider permutations of sequences of partitions, obtaining a result which paral... more In this paper we consider permutations of sequences of partitions, obtaining a result which parallels von Neumann's theorem on permutations of dense sequences and uniformly distributed sequences of points.
In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed s... more In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed sequences of partitions. Among them we found a large class having low discrepancy (which means of order 1 N). One of them is the Kakutani-Fibonacci sequence.
In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the i... more In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the inverse of the golden ratio (which we call the Kakutani–Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call the Kakutani–Fibonacci transformation) using the ‘cutting–stacking’ technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call the Kakutani–Fibonacci sequence of points), which has also been considered by other authors.
In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the i... more In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the inverse of the golden ratio (which we call the Kakutani–Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call the Kakutani–Fibonacci transformation) using the ‘cutting–stacking’ technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call the Kakutani–Fibonacci sequence of points), which has also been considered by other authors.
This paper deals with some Feller semigroups acting on a particular weighted function space on [0... more This paper deals with some Feller semigroups acting on a particular weighted function space on [0; +1[ whose generators are degenerate elliptic second order differential operators. We show that these semigroups are the transition semigroups associated with suitable Markov processes on [0; +1]. Furthermore, by means of a sequence of discrete-type positive operators we introduced in a previous paper, we evaluate the expected value and the variance of the random variables describing the position of the processes and we give an approximation formula (in the weak topology) of the distribution of the position of the processes at every time, provided the distribution of the initial position is given and possesses finite moment of order two.
Rendiconti dell'Istituto di Matematica dell'Universita di Trieste
In this paper we will generalize to higher dimension the splitting procedure introduced by Kakuta... more In this paper we will generalize to higher dimension the splitting procedure introduced by Kakutani for [0, 1]. This method will provide a sequence of nodes belonging to [0, 1] d which is uniformly distributed. The advantage of this approach is that it is intrinsecally d-dimensional.
In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed s... more In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed sequences of partitions. Among them we found a large class having low discrepancy (which means of order 1 N). One of them is the Kakutani-Fibonacci sequence.
We define a countable family of sequences of points in the unit square: the LS-sequences of point... more We define a countable family of sequences of points in the unit square: the LS-sequences of pointsà la Halton. They reveal a very strange and interesting behaviour, as well as resonance phenomena, for which we have not found an explanation, so far. We conclude with three open problems.
The LS-sequences of points recently introduced by the author are a generalization of van der Corp... more The LS-sequences of points recently introduced by the author are a generalization of van der Corput sequences. They were constructed by reordering the points of the corresponding LS-sequences of partitions. Here we present another algorithm which coincides with the classical one for van der Corput sequences and is simpler to compute than the original construction. This algorithm is based on the representation of natural numbers in base L + S and gives the van der Corput sequence in base b if L = b and S = 0. In this construction, as well as in the van der Corput one, it is essential the inversion of digits of the representation in base L + S: in this paper we also give a nice geometrical explanation of this "magical" operation.
Rendiconti del Circolo Matematico di Palermo, 2011
In this paper we consider permutations of sequences of partitions, obtaining a result which paral... more In this paper we consider permutations of sequences of partitions, obtaining a result which parallels von Neumann's theorem on permutations of dense sequences and uniformly distributed sequences of points.
In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed s... more In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed sequences of partitions. Among them we found a large class having low discrepancy (which means of order 1 N). One of them is the Kakutani-Fibonacci sequence.
In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the i... more In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the inverse of the golden ratio (which we call the Kakutani–Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call the Kakutani–Fibonacci transformation) using the ‘cutting–stacking’ technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call the Kakutani–Fibonacci sequence of points), which has also been considered by other authors.
In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the i... more In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the inverse of the golden ratio (which we call the Kakutani–Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call the Kakutani–Fibonacci transformation) using the ‘cutting–stacking’ technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call the Kakutani–Fibonacci sequence of points), which has also been considered by other authors.
This paper deals with some Feller semigroups acting on a particular weighted function space on [0... more This paper deals with some Feller semigroups acting on a particular weighted function space on [0; +1[ whose generators are degenerate elliptic second order differential operators. We show that these semigroups are the transition semigroups associated with suitable Markov processes on [0; +1]. Furthermore, by means of a sequence of discrete-type positive operators we introduced in a previous paper, we evaluate the expected value and the variance of the random variables describing the position of the processes and we give an approximation formula (in the weak topology) of the distribution of the position of the processes at every time, provided the distribution of the initial position is given and possesses finite moment of order two.
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Papers by I. Carbone