Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, … . Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of size ℤ × ℤ numbered along a square spiral, and an infinite single-quadrant chessboard (of size N × N) numbered along antidiagonals. We give a fairly complete solution in the first case, based on the Tribonacci word. There are connections with combinatorial games.@en

We discuss the base 3/2 representation of the natural numbers. We prove that the sum-of-digits function of the representation is a fixed point of a 2-block substitution on an infinite alphabet, and that this implies that sum-of-digits function modulo 2 of the representation is a fixed point x3/2 of a 2-block substitution on {0,1}. We prove that x3/2 is invariant for taking the binary complement, and present a list of conjectured properties of x3/2, which we think will be hard to prove. Finally, we make a comparison with a variant of the base 3/2 representation, and give a general result on p-q-block substitutions.@en

We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.@en

We characterize the entries of Hofstadter’s G-sequence in terms of the lower and upper Wythoff sequences. This can be used to give a short and comprehensive proof of the equality of Hofstadter’s G-sequence and the sequence of averages of the swapped Wythoff sequences. In the second part we give some results that hold when one re-places the golden mean by other quadratic algebraic numbers. In the third part we prove a close relationship between Hofstadter’s G-sequence and a sequence studied by Avdivpahić and Zejnulahi.@en

We consider in general two-block substitutions and their fixed points. We prove that some of them have a simple structure: their fixed points are morphic sequences. Others are intrinsically more complex, such as the Kolakoski sequence. We prove this for the Thue-Morse sequence in base 3/2.@en

Morphic words are letter-to-letter images of fixed points x of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of x. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called AA sequences, where α is a quadratic irrational, A is the Beatty sequence defined by A(n)=⌊αn⌋, and AA is the sequence (A(A(n))). The second example considers homomorphic embeddings of the Fibonacci language into the integers, which turns out to lead to generalized Beatty sequences with terms of the form V(n)=p⌊αn⌋+qn+r, where p,q and r are integers.@en

Let S be a map from a language Lto the integers satisfying S(vw) =S(v) +S(w)for all v, w ∈L. The classical Frobenius problem asks whether the complement of S(L)in the natural numbers will be infinite or finite, and in the latter case the value of the largest element in this complement. This is also known as the ‘coin-problem’, and Lis the full language consisting of all words over a finite alphabet. We solve the Frobenius problem for the golden mean language, any Sturmian language and the Thue–Morse language. We also consider two-dimensional embeddings. @en

This paper is an extension of the two-dimensional coupled Markov chain model developed by Elfeki and Dekking (2001) supplemented with extensive simulations. We focus on the development of various coupled Markov chains models: the so-called fully forward Markov chain, fully backward Markov chain and forward¿backward Markov chain models. We addressed many issues such as: sensitivity analysis of optimal sampling intervals in horizontal and lateral directions, directional dependency, use of Walther¿s law to describe lateral variability, effect of conditioning on number of boreholes on the model performance, stability of the Monte Carlo realizations, various implementation strategies, use of cross validation techniques to evaluate model performance and image division for statistically non-homogeneous deposits are addressed. The applications are made on three sites; two sites are located in the Netherlands, and the third is in the USA. The purpose of these applications is to show under which conditions the Markov models can be used, and to provide some guidelines for the practice. Entropy maps are good tools to indicate places where high uncertainty is present, so can be used for designing sampling networks to reduce uncertainty at these locations. Symmetric and diagonally dominant horizontal transition probabilities with proper sampling interval show plausible results (fits with geologists prediction) in terms of delineation of subsurface heterogeneous structures. Walther¿s law can be utilised with a proper sampling interval to account for the lateral variability. Keywords Markov chains - Walthers¿ law - conditional simulations - entropy - stochastic modelling - subsurface heterogeneity@en

This paper is an extension of the two-dimensional coupled Markov chain model developed by Elfeki and Dekking (2001) supplemented with extensive simulations. We focus on the development of various coupled Markov chains models: the so-called fully forward Markov chain, fully backward Markov chain and forward¿backward Markov chain models. We addressed many issues such as: sensitivity analysis of optimal sampling intervals in horizontal and lateral directions, directional dependency, use of Walther¿s law to describe lateral variability, effect of conditioning on number of boreholes on the model performance, stability of the Monte Carlo realizations, various implementation strategies, use of cross validation techniques to evaluate model performance and image division for statistically non-homogeneous deposits are addressed. The applications are made on three sites; two sites are located in the Netherlands, and the third is in the USA. The purpose of these applications is to show under which conditions the Markov models can be used, and to provide some guidelines for the practice. Entropy maps are good tools to indicate places where high uncertainty is present, so can be used for designing sampling networks to reduce uncertainty at these locations. Symmetric and diagonally dominant horizontal transition probabilities with proper sampling interval show plausible results (fits with geologists prediction) in terms of delineation of subsurface heterogeneous structures. Walther¿s law can be utilised with a proper sampling interval to account for the lateral variability. Keywords Markov chains - Walthers¿ law - conditional simulations - entropy - stochastic modelling - subsurface heterogeneity@en

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.@en

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.@en

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte¿Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte¿Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills. Keywords: Detection probability; Groundwater monitoring; Landfill; Contaminant plume; Monte¿Carlo analysis@en

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte¿Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte¿Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills. Keywords: Detection probability; Groundwater monitoring; Landfill; Contaminant plume; Monte¿Carlo analysis@en

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte¿Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte¿Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills. Keywords: Detection probability; Groundwater monitoring; Landfill; Contaminant plume; Monte¿Carlo analysis@en

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte¿Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte¿Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills. Keywords: Detection probability; Groundwater monitoring; Landfill; Contaminant plume; Monte¿Carlo analysis@en

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte¿Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte¿Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills. Keywords: Detection probability; Groundwater monitoring; Landfill; Contaminant plume; Monte¿Carlo analysis@en

Landfills represent a significant threat to groundwater contamination due to their nature of operation and their abundance. Monitoring well networks at these sites are of vital importance in detecting leakage plumes. This study presents a reliability assessment to estimate the performance of groundwater monitoring systems at landfill sites. A hypothetical problem is presented where the detection probability of several monitoring systems is compared. A Monte¿Carlo approach is used to incorporate uncertainties due to subsurface heterogeneity and the leak location. Hydraulic conductivity and leak location are considered as random variables with prescribed probability density functions. A finite difference groundwater model coupled with a random walk particle-tracking model simulates a contaminant plume released from the landfill for each Monte¿Carlo realization. The analysis shows that lateral dispersivity of the medium has a significant influence on the reliability of the monitoring system, since it is the primary parameter controlling the width of the contaminant plume. Furthermore the number and the location of the monitoring wells are dependent on the heterogeneity of the medium and size of the contaminant leak. It is concluded that the reliability of the common practice of three downgradient monitoring wells is inadequate from the point of view of prevention of groundwater contamination due to landfills. Keywords: Detection probability; Groundwater monitoring; Landfill; Contaminant plume; Monte¿Carlo analysis@en

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure.@en

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure.@en

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure.@en