18,358 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
Words with the Maximum Number of Abelian Squares
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain distinct factors that
are abelian squares. We study infinite words such that the number of abelian
square factors of length grows quadratically with .Comment: To appear in the proceedings of WORDS 201
Dvoretzky type theorems for multivariate polynomials and sections of convex bodies
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type
theorem) for homogeneous polynomials on , and improve bounds on
the number in the analogous conjecture for odd degrees (this case
is known as the Birch theorem) and complex polynomials. We also consider a
stronger conjecture on the homogeneous polynomial fields in the canonical
bundle over real and complex Grassmannians. This conjecture is much stronger
and false in general, but it is proved in the cases of (for 's of
certain type), odd , and the complex Grassmannian (for odd and even and
any ). Corollaries for the John ellipsoid of projections or sections of a
convex body are deduced from the case of the polynomial field conjecture
Unambiguous 1-Uniform Morphisms
A morphism h is unambiguous with respect to a word w if there is no other
morphism g that maps w to the same image as h. In the present paper we study
the question of whether, for any given word, there exists an unambiguous
1-uniform morphism, i.e., a morphism that maps every letter in the word to an
image of length 1.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
On generic and maximal k-ranks of binary forms
In what follows, we pose two general conjectures about decompositions of
homogeneous polynomials as sums of powers. The first one (suggested by G.
Ottaviani) deals with the generic k-rank of complex-valued forms of any degree
divisible by k in any number of variables. The second one (by the fourth
author) deals with the maximal k-rank of binary forms. We settle the first
conjecture in the cases of two variables and the second in the first
non-trivial case of the 3-rd powers of quadratic binary forms.Comment: 17 pages, 1 figur
Skip-Sliding Window Codes
Constrained coding is used widely in digital communication and storage
systems. In this paper, we study a generalized sliding window constraint called
the skip-sliding window. A skip-sliding window (SSW) code is defined in terms
of the length of a sliding window, skip length , and cost constraint
in each sliding window. Each valid codeword of length is determined by
windows of length where window starts at th symbol for
all non-negative integers such that ; and the cost constraint
in each window must be satisfied. In this work, two methods are given to
enumerate the size of SSW codes and further refinements are made to reduce the
enumeration complexity. Using the proposed enumeration methods, the noiseless
capacity of binary SSW codes is determined and observations such as greater
capacity than other classes of codes are made. Moreover, some noisy capacity
bounds are given. SSW coding constraints arise in various applications
including simultaneous energy and information transfer.Comment: 28 pages, 11 figure
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