7 research outputs found
Transition Property For Cube-Free Words
We study cube-free words over arbitrary non-unary finite alphabets and prove
the following structural property: for every pair of -ary cube-free
words, if can be infinitely extended to the right and can be infinitely
extended to the left respecting the cube-freeness property, then there exists a
"transition" word over the same alphabet such that is cube free. The
crucial case is the case of the binary alphabet, analyzed in the central part
of the paper.
The obtained "transition property", together with the developed technique,
allowed us to solve cube-free versions of three old open problems by Restivo
and Salemi. Besides, it has some further implications for combinatorics on
words; e.g., it implies the existence of infinite cube-free words of very big
subword (factor) complexity.Comment: 14 pages, 5 figure
Branching densities of cube-free and square-free words
Binary cube-free language and ternary square-free language are two “canonical” represen-tatives of a wide class of languages defined by avoidance properties. Each of these two languages can be viewed as an infinite binary tree reflecting the prefix order of its elements. We study how “homogenious” these trees are, analysing the following parameter: the density of branching nodes along infinite paths. We present combinatorial results and an efficient search algorithm, which together allowed us to get the following numerical results for the cube-free language: the minimal density of branching points is between 3509/9120 ≈ 0.38476 and 13/29 ≈ 0.44828, and the maximal density is between 0.72 and 67/93 ≈ 0.72043. We also prove the lower bound 223/868 ≈ 0.25691 on the density of branching points in the tree of the ternary square-free language. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.This research was funded by Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2020-1537/1)
Subword complexity and power avoidance
We begin a systematic study of the relations between subword complexity of
infinite words and their power avoidance. Among other things, we show that
-- the Thue-Morse word has the minimum possible subword complexity over all
overlap-free binary words and all -power-free binary words, but not
over all -power-free binary words;
-- the twisted Thue-Morse word has the maximum possible subword complexity
over all overlap-free binary words, but no word has the maximum subword
complexity over all -power-free binary words;
-- if some word attains the minimum possible subword complexity over all
square-free ternary words, then one such word is the ternary Thue word;
-- the recently constructed 1-2-bonacci word has the minimum possible subword
complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC
Subword complexity and power avoidance
We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that – the Thue–Morse word has the minimum possible subword complexity over all overlap-free binary words and all ( [Formula presented] )-power-free binary words, but not over all ( [Formula presented] )+-power-free binary words; – the twisted Thue–Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all ( [Formula presented] )-power-free binary words; – if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; – the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all symmetric square-free ternary words. © 2018 Elsevier B.V
Constructing Premaximal Binary Cube-free Words of Any Level
We study the structure of the language of binary cube-free words. Namely, we are interested in the cube-free words that cannot be infinitely extended preserving cube-freeness. We show the existence of such words with arbitrarily long finite extensions, both to one side and to both sides
XVIII International Coal Preparation Congress
Changes in economic and market conditions of mineral raw materials in recent
years have greatly increased demands on the ef fi ciency of mining production. This
is certainly true of the coal industry. World coal consumption is growing faster than
other types of fuel and in the past year it exceeded 7.6 billion tons. Coal extraction
and processing technology are continuously evolving, becoming more economical
and environmentally friendly. “ Clean coal ” technology is becoming increasingly
popular. Coal chemistry, production of new materials and pharmacology are now
added to the traditional use areas — power industry and metallurgy. The leading role
in the development of new areas of coal use belongs to preparation technology and
advanced coal processing. Hi-tech modern technology and the increasing interna-
tional demand for its effectiveness and ef fi ciency put completely new goals for the
University. Our main task is to develop a new generation of workforce capacity and
research in line with global trends in the development of science and technology to
address critical industry issues.
Today Russia, like the rest of the world faces rapid and profound changes
affecting all spheres of life. The de fi ning feature of modern era has been a rapid
development of high technology, intellectual capital being its main asset and
resource. The dynamics of scienti fi c and technological development requires acti-
vation of University research activities. The University must be a generator of ideas
to meet the needs of the economy and national development. Due to the high
intellectual potential, University expert mission becomes more and more called for
and is capable of providing professional assessment and building science-based
predictions in various fi elds.
Coal industry, as well as the whole fuel and energy sector of the global economy
is growing fast. Global multinational energy companies are less likely to be under
state in fl uence and will soon become the main mechanism for the rapid spread of
technologies based on new knowledge. Mineral resources will have an even greater
impact on the stability of the economies of many countries. Current progress in the
technology of coal-based gas synthesis is not just a change in the traditional energy markets, but the emergence of new products of direct consumption, obtained from
coal, such as synthetic fuels, chemicals and agrochemical products. All this requires
a revision of the value of coal in the modern world economy