Generation and Recognition of Digital Planes Using Multi-dimensional Continued Fractions

In his paper "Multidimensional continued fractions" ( Ann. Univ. Sci. Budapest. Eötvös Sect. Math. , v. 13, 1970, pp. 113-140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each k -tuple ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) of real numbers (satisfying 0 > α i > 1 0 > {\alpha _i} > 1 ) a sequence b 1 , b 2 , … {b_1}, {b_2}, \ldots of positive integers; this sequence is called a continued k -fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for α 1 {\alpha _1} . A simple recursive procedure applied to b 1 , b 2 , … {b_1}, {b_2}, \ldots produces a sequence a ( n ) = ( A n ( 1 ) / B n , … , A n ( k ) / B n ) ( n = 1 , 2 , … ; A n ( i ) ⩾ 0 a(n) = (A_n^{(1)}/{B_n}, \ldots ,A_n^{(k)}/{B_n})\;(n = 1,2, \ldots ;A_n^{(i)} \geqslant 0 and B n > 0 {B_n} > 0 are integers) of simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and a sequence c ( n ) = ( c n 0 , c n 1 , … , c n k ) ( n = 1 , 2 , … ) c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots ) of integer ( k + 1 ) (k + 1) -tuples such that the linear combination c n 0 + c n 1 α 1 + ⋯ + c n k α k {c_{n0}} + {c_{n1}}{\alpha _1} + \cdots + {c_{nk}}{\alpha _k} approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence a ( 1 ) , a ( 2 ) , … a(1),a(2), \ldots contains all of the "best" simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and that the sequence c ( 1 ) , c ( 2 ) , … c(1),c(2), \ldots contains all of the "best" approximations to zero by the linear form x 0 + x 1 α 1 + ⋯ + x n α n {x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n} . For the special case k = 2 and α 1 = θ 2 − 1 , α 2 = θ − 1 {\alpha _1} = {\theta ^2} - 1,{\alpha _2} = \theta - 1 (where θ = 2 cos ⁡ ( 2 π / 7 ) \theta = 2\cos (2\pi /7) is the positive root of x 3 + x 2 − 2 x − 1 = 0 {x^3} + {x^2} - 2x - 1 = 0 , Szekeres further conjectured that the 2-fraction b 1 , b 2 , … {b_1},{b_2}, \ldots is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form x 0 + x 1 ( θ 2 − 1 ) + x 2 ( θ − 1 ) {x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1) and concerning almost periodicity for the 2-fraction of ( θ 2 − 1 , θ − 1 ) ({\theta ^2} - 1,\theta - 1) are proved. The method used can be applied to other pairs of cubic irrationals α 1 , α 2 {\alpha _1},{\alpha _2} .

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

A simplex–karyon algorithm for expanding real numbers α = (α1,..., α d ) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ℘(r)-strategy for constructing convergents is chosen, with a free objective function ℘(r) on the hyperplane K.

The problem of simultaneously approximating a vector of irrational numbers with rationals is analyzed in a geometrical setting using notions of dynamical systems theory. We discuss here a (vectorial) multidimensional continued-fraction algorithm (MCFA) of additive type, the generalized mediant algorithm (GMA), and give a geometrical interpretation to it. We calculate the invariant measure of the GMA shift as well as its Kolmogorov-Sinai (KS) entropy for arbitrary number of irrationals. The KS entropy is related to the growth rate of denominators of the Euclidean algorithm. This is the first analytical calculation of the growth rate of denominators for any MCFA.

In the first part of this book we saw that the LLS sequences completely determine all possible sails of integer angles in the one-dimensional case. The situation in the multidimensional case is much more complicated. Of course, the convex hull algorithms can compute all the vertices and faces of sails for finite continued fractions, but it is not clear how to construct (or to describe) vertices of infinite sails of multidimensional continued fractions in general. What integer-combinatorial structures could the infinite sails have? There is no single example in the case of aperiodic infinite continued fractions of dimension greater than one. The situation is better with periodic algebraic sails, where each sail is characterized by its fundamental domain and the group of period shifts (i.e., the positive Dirichlet group).

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

Schweiger introduced the notion of a subtractive algorithm, to classify certain types of multidimensional continued fractions. We study the limit behaviour of one particular subtractive algorithm, which generalises a continued fraction algorithm that was originally proposed by Selmer. The algorithm that we study depends on two parameters a and b. We first find a Markov partition if a ≥ b. Using inducing techniques, we then prove the existence of an ergodic absolutely continuous invariant probability measure a ≥ b for the quality of the rational approximations for Lebesgue-typical multidimensional vectors.

V. Berthe showed that the frequencies of factors in a Sturmian word of slope α, as well as the number of factors with a given frequency, can be expressed in terms of the continued fraction expansion of α. In this paper we describe a multi-dimensional continued fraction process associated with a class of sequences of (block) complexity kn+1 originally introduced by P. Arnoux and G. Rauzy. This vectorial division algorithm yields simultaneous rational approximations of the frequencies of the letters. We extend Berthe’s result to factors of Arnoux-Rauzy sequences by expressing both the frequencies and the number of factors with a given frequency, in terms of the ‘convergents’ obtained from the generalized continued fraction expansion of the frequencies of the letters.

The Guting algorithm for constructing multidimensional continued fractions is considered. It is proved that, in the case of dimension 2, this algorithm can be used to find the coefficients of the linear dependence of numbers; a criterion is given for verifying that the partial quotients furnished by the algorithmare, indeed, elements of the continued fraction for the expanded (generally irrational) numbers.

Digital polynomial curves and surfaces arise from the digitization of algebraic surfaces such as lines, parabolas, planes, or paraboloids. It is known that the n-th difference of a digital polynomial curve of degree n is periodic for polynomials that have rational coefficients. In this paper we consider the following problem: Suppose we have a digital curve S whose n-th difference is known to be periodic. When is S a digital rational polynomial curve? As a solution to this problem we state a simple criterion that can be checked in linear time. As a first application of this criterion we describe a linear time algorithm for the recognition of digital straight lines. In comparison to other algorithms, the advantages of the new algorithm are its simplicity, and its ability to actually find the coefficients of the rational polynomial representing the line. We then go on to discuss the applicability of this criterion to the recognition of digital curves and surfaces of arbitrary degree.

The rapid development of industrial automation systems has generated massive industrial data volumes. To address the complexity of big data processing and analysis in automation systems, this research proposes a multi-dimensional data analysis framework based on distributed computing architecture. The framework implements an enhanced MapReduce algorithm for heterogeneous data processing and performs in-depth analysis of industrial process data through sequential stages of data preprocessing, feature extraction, and pattern recognition. Experimental validation demonstrates that this framework significantly improves data processing efficiency, achieving 95.8% accuracy with an 18.3% enhancement over conventional methodologies. The framework exhibits notable advantages in application scenarios including equipment failure prediction and production quality control, thereby providing technical foundation for intelligent upgrades of automation systems.

Assume that a black/white n×n image is stored one pixel per element of a vector of length n2. We consider determining some characteristics of such images using the scan model of parallel computation. The scan model was introduced by G.E. Blelloch and is a single instruction multiple data (SIMD) vector model of computation. The primitive operations of the model work on vectors (one dimensional arrays) of values, with three types of primitive operations: Elementwise arithmetic and logical operations, permutation operations, and scan operations, a type of prefix computation (a scan operation takes a binary associative operator ⊗ and a vector [a1,…, an] and returns the vector [a1, a1⊗a2,…, a1⊗a2⊗…⊗an]). We show that many important characteristics of binary images can be determined in constant time on the scan model of parallel computation. These include the convex hull construction, diameter, width, smallest enclosing box, perimeter, area, detecting digital convexity, parallel and point visibility (determining for each pixel of the image the portion that is visible, i.e., not obstructed by any black pixel, in a given direction from infinity or from a given point, respectively) of an image, smallest, largest and Hausdorff distances between two images, linear separability of two images, and the recognition of digital lines, rectangles and arcs.

This paper investigates the properties of digital hyperplanes of arbitrary dimension. We extend previous results that have been obtained for digital straight lines and digital planes, namely, Hung's evenness, Rosenfeld's chord, and Kim's chordal triangle property. To characterize digital hyperplanes we introduce the notion of digital flatness. We make a distinction between flatness and local flatness. The main tool we use is Helly's First Theorem, a classical result on convex sets, by means of which precise and verifiable conditions are given for the flatness of digital point sets. The main result is the proof of the equivalence of local flatness, evenness, and the chord property for certain infinite digital point sets in spaces of arbitrary dimension.

Leader-follower dynamics play a fundamental role in coordinated interactions between two or more subjects. Due to the complexity of social interactions, they can emerge spontaneously in social settings even without an explicit role assignment. Investigating these dynamics at the neural level requires techniques capable of capturing inter-brain coordination, such as hyperscanning, which enables the simultaneous recording of brain activity from multiple subjects. While some studies have used the direction of the causal relations in the inter-brain model as an indication of the roles assumed, no standardized method has yet been developed to identify leader and follower roles as they emerge naturally during an interaction. In this study, we link the results of an alignment method based on multi-dimensional network framework with behavioral data collected from 32 neurotypical volunteers during a Joint Action task in an EEG-based hyperscanning setting. By analyzing multiple-subject networks with multi-dimensional graph theory, we quantify differences in inter-brain connectivity and derive a measure, the interimbalance index, that reflects role asymmetries within each dyad. These asymmetries were characterized as indicative of a leader-follower relation by means of a statistical comparison with a behavioral measure of leadership tendency. Although future work will be needed to further characterize the subjects' roles from behavioral measure, our results shows how the proposed index effectively realigns subjects according to their roles.Clinical Relevance- This study lays the groundwork for investigating leader-follower dynamics at the neural level, enabling the detection of potential atypical patterns in subjects with altered social behavior.

Ergodicity for [formula omitted]-adic continued fraction algorithms