A Note on the Abelian Complexity of the Rudin-Shapiro Sequence

Let {r(n)}n≥0 be the Rudin-Shapiro sequence, and let ρ(n):=max{∑j=ii+n−1r(j)∣i≥0}+1 be the abelian complexity function of the Rudin-Shapiro sequence. In this note, we show that the function ρ(n) has many similarities with the classical summatory function Sr(n):=∑i=0nr(i). In particular, we prove that for every positive integer n, 3≤ρ(n)n≤3. Moreover, the point set {ρ(n)n:n≥1} is dense in [3,3].

Implementing Stylistics in IELTS Writing Preparation Focusing on Noun Phrases

In the IELTS writing preparation class, the teacher can facilitate the students by integrating the stylistic approach in building writing students’ style. Not only achieving the primary goals in English proficiency by focusing on the writing style, but students can also comfortably share their ideas in their style. The paper examined the stylistics integrated material in IELTS writing through noun phrase construction. This study employed a qualitative method and a descriptive design. The data were collected using structured observation sheets as the instrument and were analyzed stylistically. Writing materials and the writing results of 12 students at an English course were collected. Results showed that stylistics was used in phrase complexity and ambiguity as a part of writing strategies. It was supported by the three dominant types of noun phrases produced by the students in their writing: determiner + head, determiner + pre-modifier + head, and pre-modifier + head, as the evidence of the implementation of integrated material in IELTS writing. For the noun phrase uses, the students could vary the noun phrases in their complexity, function, and type in their tests, though the variations in each of those aspects were still low. Implications of the results are stylistics could be implemented in language learning-teaching activities in general. It could encourage students to develop their writing style for self-expression and more explicit writing products.

Subsystems of transitive subshifts with linear complexity

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].

Subshifts on Infinite Alphabets and Their Entropy

We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.

RESEARCH OF RADIO FREQUENCY SPECTROGRAMS USING METHODS OF LINGUISTIC ANALYSIS

The article presents the study of experimental curves of the spectral spectra of radio waves of the FM-range developed on a laboratory model. The algorithm was used to define sections, for which the complexity function took locally minimal values. The standard was determined for each section of the curve, in which all the arithmetic mean ordinates of all areas correspond to the certain class. For a more extended linguistic description of the experimental curves, it is proposed to compile the description taking into account the location of the curve sections on the abscissa axis. The obtained extended linguistic description of the curve will reflect not only classes of simple events, but their phases as well. As a result of applying the linguistic analysis system for the analysis of spectral radiograms, it can be inferred that the experimental curves are presented in the form of short and reliable rules for the analysis of the radiogram spectrum. The use of standards allows to accurately represent each chain of characters in each group with a minimum distance to the standard. The obtained extended descriptions quite accurately describe the behavior of the curves studied.

Cut and project sets with polytopal window I: Complexity

We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

QUANTILE-BASED APPROACH TO ESTIMATING COGNITIVE TEXT COMPLEXITY

This paper introduces an approach to measuring the cognitive complexity of texts on various language levels. While standard readability indices are based on the linear combination of primary statistics, our general approach allows us to estimate complexity on morphological, lexical, syntactic, and discursive levels. Each model is defined by the tokens for the specific language level and the complexity function of a single token. We then use the reference collection of moderately complex texts and the quantile-based approach to spot the abnormally rare tokens. The proposed supervised ensemble, based on the ElasticNet model, incorporates models from all language levels. Having collected a labeled dataset through crowdsourcing, consisting of pairs of articles from the Russian Wikipedia, we consider several models and ensembles and compare them to common baselines. Suggested models are flexible due to the freedom in choosing the reference collection. The described experiments confirm the competitiveness of the proposed approach, as the ensembles demonstrate the best target metric value.