66 research outputs found
Sums of Digits, Overlaps, and Palindromes
Let s_k(n) denote the sum of the digits in the base-k representation of n. In a celebrated paper, Thue showed that the infinite word (s_2(n) \bmod 2)_n≥ 0 is \emphoverlap-free, i.e., contains no subword of the form axaxa where x is any finite word and a is a single symbol. Let k,m be integers with k>2, m≥ 1. In this paper, generalizing Thue's result, we prove that the infinite word t_k,m := (s_k(n) \bmod m)_n≥ 0 is overlap-free if and only if m≥ k. We also prove that t_k,m contains arbitrarily long squares (i.e., subwords of the form xx where x is nonempty), and contains arbitrarily long palindromes if and only if m≤ 2
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
Sequences of numbers obtained by digit and iterative digit sums of Sophie Germain primes and its variants
Sequences were generated by the digit and iterative digit sums of Sophie
Germain and Safe primes and their variants. The results of the digit and
iterative digit sum of Sophie Germain and Safe primes were almost the same.
The same applied to the square and cube of the respective primes. Also, the
results of the digit and iterative digit sum of primes that are not Sophie
Germain are the same with the primes that are notSafe. The results of the digit
and iterative digit sum of prime that are either Sophie Germain or Safe are like
the combination of the results of the respective primes when considered
separatel
Constructions of words rich in palindromes and pseudopalindromes
A narrow connection between infinite binary words rich in classical
palindromes and infinite binary words rich simultaneously in palindromes and
pseudopalindromes (the so-called -rich words) is demonstrated.
The correspondence between rich and -rich words is based on the operation
acting over words over the alphabet and defined by
, where .
The operation enables us to construct a new class of rich words and a new
class of -rich words.
Finally, the operation is considered on the multiliteral alphabet
as well and applied to the generalized Thue--Morse words. As a
byproduct, new binary rich and -rich words are obtained by application of
on the generalized Thue--Morse words over the alphabet .Comment: 26 page
Doppler Tolerance, Complementary Code Sets and the Generalized Thue-Morse Sequence
We generalize the construction of Doppler-tolerant Golay complementary
waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having
more than two codes. This is accomplished by exploiting number-theoretic
results involving the sum-of-digits function, equal sums of like powers, and a
generalization to more than two symbols of the classical two-symbol
Prouhet-Thue-Morse sequence.Comment: 12 page
On the critical exponent of generalized Thue-Morse words
For certain generalized Thue-Morse words t, we compute the "critical
exponent", i.e., the supremum of the set of rational numbers that are exponents
of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer
Science (accepted October 15, 2007
Latin Square Thue-Morse Sequences are Overlap-Free
We define a morphism based upon a Latin square that generalizes the
Thue-Morse morphism. We prove that fixed points of this morphism are
overlap-free sequences generalizing results of Allouche - Shallit and Frid.Comment: 5 pages, 1 figur
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