A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

2013 ◽  
Vol 23 (04) ◽  
pp. 963-987 ◽  
Author(s):  
FILIPPO MIGNOSI ◽  
ANTONIO RESTIVO
Keyword(s):  
The Other ◽  
Factorization Theorem ◽  
Complexity Measures ◽  
Infinite Words ◽  
Average Value ◽  
Complexity Function ◽  
Binary Word ◽  
Infinite Word

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

scholarly journals Entropy ratio for infinite sequences with positive entropy

2018 ◽  
Vol 40 (3) ◽  
pp. 751-762 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Exponential Growth Rate ◽  
Positive Entropy ◽  
Infinite Words ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of word entropy$E_{W}(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_{W}(f)$ in terms of the limiting lower exponential growth rate of $f$.

scholarly journals WORD COMPLEXITY AND REPETITIONS IN WORDS

2004 ◽  
Vol 15 (01) ◽  
pp. 41-55 ◽  
Author(s):  
LUCIAN ILIE ◽  
SHENG YU ◽  
KAIZHONG ZHANG
Keyword(s):  
Data Compression ◽  
Fast Algorithms ◽  
Complexity Measure ◽  
Combinatorics On Words ◽  
Complexity Measures ◽  
Infinite Words ◽  
Infinite Word ◽  
De Bruijn ◽  
Subword Complexity

With ideas from data compression and combinatorics on words, we introduce a complexity measure for words, called repetition complexity, which quantifies the amount of repetition in a word. The repetition complexity of w, R (w), is defined as the smallest amount of space needed to store w when reduced by repeatedly applying the following procedure: n consecutive occurrences uu…u of the same subword u of w are stored as (u,n). The repetition complexity has interesting relations with well-known complexity measures, such as subword complexity, SUB , and Lempel-Ziv complexity, LZ . We have always R (w)≥ LZ (w) and could even be that the former is linear while the latter is only logarithmic; e.g., this happens for prefixes of certain infinite words obtained by iterated morphisms. An infinite word α being ultimately periodic is equivalent to: (i) [Formula: see text], (ii) [Formula: see text], and (iii) [Formula: see text]. De Bruijn words, well known for their high subword complexity, are shown to have almost highest repetition complexity; the precise complexity remains open. R (w) can be computed in time [Formula: see text] and it is open, and probably very difficult, to find fast algorithms.

scholarly journals Complexity and fractal dimensions for infinite sequences with positive entropy

2019 ◽  
Vol 21 (06) ◽  
pp. 1850068
Author(s):  
Christian Mauduit ◽  
Carlos Gustavo Moreira
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Combinatorial Proof ◽  
Infinite Words ◽  
Entropy Formula ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word [Formula: see text] on a finite alphabet [Formula: see text] is the sequence counting, for each non-negative [Formula: see text], the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of the infinite word [Formula: see text]. The goal of this work is to estimate the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of an infinite word [Formula: see text] with a complexity function bounded by a given function [Formula: see text] with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy [Formula: see text] associated to a given function [Formula: see text] and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by [Formula: see text] in terms of its word entropy. We present a combinatorial proof of the fact that [Formula: see text] is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by [Formula: see text] and we give several examples showing that even under strong conditions on [Formula: see text], the word entropy [Formula: see text] can be strictly smaller than the limiting lower exponential growth rate of [Formula: see text].

ON SHALLOW‐HOLE TEMPERATURE MEASUREMENTS—A TEST STUDY IN THE SALTON SEA GEOTHERMAL FIELD

Geophysics ◽  
1977 ◽  
Vol 42 (3) ◽  
pp. 572-583 ◽  
Author(s):  
Tien‐Chang Lee
Keyword(s):  
Numerical Methods ◽  
Ground Surface ◽  
Nonlinear Least Squares ◽  
Geothermal Gradient ◽  
Geothermal Field ◽  
The Other ◽  
Temperature Measurements ◽  
Salton Sea ◽  
Depth Interval ◽  
Average Value

Shallow‐hole (<13 m) temperature measurements made at various depths and/or times may yield reliable values of geothermal gradient and thermal diffusivity if the groundwater table is shallow (a few meters) such that the effect of time‐dependent moisture content and physical properties is negligible. Two numerical methods based on nonlinear least‐squares curve fitting are derived to remove the effect of annual temperature wave at the ground surface. One method can provide information on the gradient and diffusivity as a function of depth while the other gives average value over the depth interval measured. Experiments were carried in six test holes cased with 2 cm OD PVC pipes in the Salton Sea geothermal field. A set of 5 to 7 thermistors was permanently buried inside the individual pipes with dry sand. Consistent gradient determinations have been obtained with both numerical methods from six monthly observations. By linearly extrapolating the depths to the 100°C and 200°C isotherms from the calculated gradients and mean ground temperatures, we have found good agreement with the nearby deep‐well data for four holes. Discrepancy is found for two holes, one of which is located near the field of [Formula: see text] mud volcanoes and the other near the volcanic Red Hill, reflecting complicated local hydrologic conditions.

scholarly journals On Newcomb's method of investigating periodicities and its application to Brückner's weather cycle

1914 ◽  
Vol 90 (619) ◽  
pp. 349-355
Keyword(s):  
Thermal Radiation ◽  
The Other ◽  
New Method ◽  
Critical Examination ◽  
Numerical Work ◽  
Average Value ◽  
Other Hand ◽  
Relationship Of ◽  
The Relationship ◽  
Great Simplicity

During the last few years of his life Prof. Simon Newcomb was keenly interested in the problem of periodicities, and devised a new method for their investigation. This method is explained, and to some extent applied, in a paper entitled "A Search for Fluctuations in the Sun's Thermal Radiation through their Influence on Terrestrial Temperature." The importance of the question justifies a critical examination of the relationship of the older methods to that of Newcomb, and though I do not agree with his contention that his process gives us more than can be obtained from Fourier's analysis, it has the advantage of great simplicity in its numerical work, and should prove useful in a certain, though I am afraid, very limited field. Let f ( t ) represent a function of a variable which we may take to be the time, and let the average value of the function be zero. Newcomb examines the sum of the series f ( t 1 ) f ( t 1 + τ) + f ( t 2 ) f ( t 2 + τ) + f ( t 3 ) f ( t 3 + τ) + ..., where t 1 , t 2 , etc., are definite values of the variable which are taken to lie at equal distances from each other. If the function be periodic so as to repeat itself after an interval τ, the products are all squares and each term is positive. If, on the other hand, the periodic time be 2τ, each product will be negative and the sum itself therefore negative. It is easy to see that if τ be varied continuously the sum of the series passes through maxima and minima, and the maxima will indicated the periodic time, or any of its multiples.

The effects of direct drilling, shallow cultivation and ploughing on some soil physical properties in a long-term experiment on spring barley

1985 ◽  
Vol 104 (1) ◽  
pp. 125-133 ◽  
Author(s):  
K. Chaney ◽  
D. R. Hodgson ◽  
M. A. Braim
Keyword(s):  
Spring Barley ◽  
The Other ◽  
Clay Loam ◽  
Average Value ◽  
Structural Problems ◽  
Physical Measurements ◽  
Long Term Experiment ◽  
Direct Drilling ◽  
Air Capacity

SummaryPhysical measurements were made on the soil of a long-term cultivation experiment comparing direct drilling, tine cultivation and mouldboard ploughing for spring barley to investigate possible reasons for differences in yield. The soil was a typical argillio brown earth, approximately 90 cm of sandy clay loam topsoil and clay loam subsoil overlying magnesian limestone. For the three periods 1971–4, 1975–7 and 1978–80 the mean grain yields were marginally lower after direct drilling than after shallow cultivation or ploughing. There was an average decline in yield of 1·33 t/ha from the first to the last period, the decline being greater for direct drilling than the other two tillage systems. Although the surface horizon (0–5 cm) of direct-drilled soil had a higher content of organic matter than the ploughed, this did not increase the stability of the aggregates. Slaking tests had shown the soil to be inherently unstable and likely to suffer from structural problems. After the first 3 years bulk density of direct-drilled soil (0–15 cm) increased markedly to ca. l·5 g/cm8 and then remained relatively stable. In the ploughed soil, density increased steadily over the period to an average value of co. 1·45 g/cm8. Tine cultivation to 7–8 cm reduced cone resistance values in the surface compared with direct-drilled soil but below 15 cm there were no significant differences. Ploughing gave significantly lower values than direct drilling to a depth of 30 cm. Measurements of pore sizes in direct-drilled and ploughed soil were highly variable with few significant differences. Mean air capacity values (1978–80) tended to be lower in direct-drilled than in ploughed topsoil particularly for plots direct drilled after 7 years of deep tine cultivation. A limited number of root measurements in 1978 and 1980 showed that the length of root per unit of ground area was much less after direct drilling than after ploughing. Shallow cultivation, surprisingly, gave most root with a greater proportion of the root system below 20 cm than in the other two treatments. The classification of this soil according to its suitability for direct drilling cereals is discussed.

scholarly journals EFFECT OF VARIATIONS OF BETA VULGARIS EXTRACTS ON MASKING THE BITTER TASTE OF MOMORDICA CHARANTIA EXTRACT SYRUP

2018 ◽  
Vol 10 (1) ◽  
pp. 159
Author(s):  
Sutriyo . ◽  
Raditya Iswandana ◽  
Elisa Nur Widiya
Keyword(s):  
Specific Gravity ◽  
Physical Properties ◽  
Beta Vulgaris ◽  
Sodium Benzoate ◽  
Bitter Taste ◽  
Momordica Charantia ◽  
The Other ◽  
Bitter Melon ◽  
Average Value

Objective: This study aimed to obtain a formula with an optimal sweetener concentration of beet extract that can cover the bitter taste of bitter melonand confer optimal physical properties on the syrup.Methods: The syrups were prepared by mixing bitter melon extract, sucrose, beet extract, sorbitol, sodium benzoate, strawberry essence, anddemineralized water. The control formula and formulas 1, 2, and 3 contained beet extract at concentrations of 0% and 10%, 15%, and 20%, respectively.All formulas were evaluated to determine their physical properties, stability, and bitterness. The bitterness was tested on 30 respondents, with databeing analyzed using Wilcoxon’s test on SPSS software.Results and Conclusion: Formula 3 with 20% beet extract was identified as the best formula for masking bitter taste because it had a significantlybetter average value than the other formulas (p<0.05) and the highest bitterless taste percentage (86.67%), with physical properties of a brownishblackcolor, odor of mixture of strawberry and dominant beet, a sweet and dominant beet taste, pH 5.46, and specific gravity of 1.228 g/mL.

scholarly journals Profiling of Individual Desulfo-Glucosinolate Content in Cabbage Head (Brassica oleracea var. capitata) Germplasm

Molecules ◽  
2020 ◽  
Vol 25 (8) ◽  
pp. 1860 ◽  
Author(s):  
Shiva Ram Bhandari ◽  
Juhee Rhee ◽  
Chang Sun Choi ◽  
Jung Su Jo ◽  
Yu Kyeong Shin ◽  
...  
Keyword(s):  
Significant Variation ◽  
The Other ◽  
Glucosinolate Content ◽  
Breeding Programs ◽  
Average Value ◽  
Hydrolysis Products ◽  
Coefficients Of Variation ◽  
Anti Cancer ◽  
The Individual ◽  
Different Origins

Individual glucosinolates (GSLs) were assessed to select cabbage genotypes for a potential breeding program. One hundred forty-six cabbage genotypes from different origins were grown in an open field from March to June 2019; the cabbage heads were used for GSL analyses. Seven aliphatics [glucoiberin (GIB), progoitrin (PRO), epi-progoitrin (EPI), sinigrin (SIN), glucoraphanin (GRA), glucoerucin (GER) and gluconapin (GNA)], one aromatic [gluconasturtiin (GNS)] and four indolyl GSLs [glucobrassicin (GBS), 4-hydroxyglucobrassicin (4HGBS), 4-methoxyglucobrassicin (4MGBS), neoglucobrassicin (NGBS)] were found this study. Significant variation was observed in the individual GSL content and in each class of GSLs among the cabbage genotypes. Aliphatic GSLs were predominant (58.5%) among the total GSLs, followed by indolyl GSL (40.7%) and aromatic GSLs (0.8%), showing 46.4, 51.2 and 137.8% coefficients of variation, respectively. GIB, GBS and NGBS were the most common GSLs found in all genotypes. GBS was the most dominant GSL, with an average value of 3.91 µmol g−1 (0.79 to 13.14 µmol g−1). SIN, GIB, PRO and GRA were the other major GSLs, showing average values of 3.45, 1.50, 0.77 and 0.62 µmol g−1, respectively. The genotypes with relatively high contents of GBS, SIN, GIB and GRA warrant detailed studies for future breeding programs since the hydrolysis products of these GSLs have several anti-cancer properties.

Esophageal and pleural pressures in man, upright and supine

1959 ◽  
Vol 14 (1) ◽  
pp. 81-83 ◽  
Author(s):  
Jere Mead ◽  
E. A. Gaensler
Keyword(s):  
Supine Position ◽  
Esophageal Pressure ◽  
The Other ◽  
Pleural Pressure ◽  
Upright Posture ◽  
Quiet Breathing ◽  
Individual Values ◽  
Average Value ◽  
Supine Posture ◽  
The Difference

To assess the reliability of esophageal sampling of pleural pressure, both pressures were recorded simultaneously in seven subjects in the upright posture and, in five of these, in the supine posture. Esophageal pressure was obtained with air-containing latex balloons and pleural pressure was recorded from catheters introduced into the intrapleural space. Pneumothorax volumes were estimated to be between 5 and 350 ml. The closest correspondence between the pressures was observed in the upright posture. Expressing the difference in the amplitude of the respiratory excursions as the percentage of deviation of esophageal from pleural pressures, in the upright posture individual values ranged from +20 to -18, while the average value for the group did not differ significantly from 0. On the other hand, in the supine position individual variation was greater (range +60 to -25) and, on the average, esophageal pressure amplitudes were larger (average +27 during quiet breathing) than pleural pressure amplitudes. Submitted on August 22, 1958

scholarly journals Transport of dissolved load in the hydrologically extreme conditions in the Crnica river basin

2012 ◽  
Vol 92 (4) ◽  
pp. 17-30
Author(s):  
Sanja Mustafic ◽  
Tanja Dobrosavljevic ◽  
Predrag Manojlovic ◽  
Milan Srejic
Keyword(s):  
The Other ◽  
Runoff Water ◽  
Water Mineralization ◽  
Average Value ◽  
Time Period ◽  
Average Water ◽  
Annual Frequency ◽  
Dissolved Load ◽  
Annual Amount

During 2010, 62065.3 t of the dissolved load were transported from the Crnica Basin, which was 5.1 times higher than the long-term average value. In hydrological terms the studied year was specific. In the period from January to May the amount of runoff water was significantly above the average perennial values, while in the period from August to November the amount was below the perennial average. According to the hydrological conditions in the period January-May 82.3% were transported, and in the period August-November only 5.1% of the annual dissolved load amount. Most of the year (54.2% of the time) the specific runoff was less than 10 l/s/km2, the average water mineralization was 374.9 mg/l, and in that time 13% of the total annual amount of the dissolved load was evacuated. On the other hand, the specific runoffs greater than 50 l/s/km2 represented 13.4% of the annual frequency, the average water mineralization was 263.3 mg/l, but for that time period even 53.9% of the annual quantity of the dissolved load was evacuated.

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