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UNAVOIDABLE SETS OF CONSTANT LENGTH

2004
*
International journal of algebra and computation
*

For k, q ≥ 1, let c(k, q) be the number

doi:10.1142/s0218196704001700
fatcat:do7ozqsadbcojh46f3limxwl2m
*of*conjugacy classes*of*words*of**length*k on q letters. An*unavoidable**set**of*words*of**length*k on q symbols has at least c(k, q) elements. ... We show that for any k, q ≥ 1 there exists an*unavoidable**set**of*words*of**length*k on q symbols having c(k, q) elements. ... It is easy to see that an*unavoidable**set**of*words*of**constant**length*k on some alphabet A has to contain at least one word*of*each conjugacy class*of*words*of**length*k on this alphabet. ...##
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Number of holes in unavoidable sets of partial words II

2012
*
Journal of Discrete Algorithms
*

Towards this, we investigate the minimum size

doi:10.1016/j.jda.2011.12.002
fatcat:cwk24hdtr5e5zkjyejt4e4jgie
*of**unavoidable**sets**of*partial words with a fixed number*of*holes. ... Additionally, we analyze the complexity*of*variations on the decision problem when placing restrictions on the number*of*holes and*length**of*the words. ... words in a*set*and returns bounds on the minimum cardinality*of*an*unavoidable**set*with those parameters. ...##
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Unavoidable sets

2006
*
Theoretical Computer Science
*

We obtain new results on minimum

doi:10.1016/j.tcs.2006.03.024
fatcat:v5uzfmcybnacvhxchb6alspw5u
*lengths**of*words in an*unavoidable**set**of*words*of*cardinality n before introducing the notion*of*aperiodic*unavoidable**sets*, a natural extension*of**unavoidable**sets*. ... Let X be an*unavoidable**set**of*non-empty words all*of**length*m + 1. Then for some v ∈ A m we have {va 1 , va 2 , . . . , va k } ⊆ X. Proof. ... Acknowledgement The authors would like to thank the referee for finding the error in Section 3*of*the original draft and essentially devising Examples 22 and 23. ...##
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Page 814 of Mathematical Reviews Vol. , Issue 86b
[page]

1986
*
Mathematical Reviews
*

The language L(p) is the

*set**of*all words over the alphabet*of**constant*symbols generated from p by substituting some nonempty words for the variables in p. ... A grammar G = (5, V, P) is said to be D-algebraic if V is a finite*set**of*nonterminals and if P = {(a > m):a€V,me€(ZUV)’} is such that the*length**of*m is bounded. ...##
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On the complexity of deciding avoidability of sets of partial words

2010
*
Theoretical Computer Science
*

We give a polynomial bound on the period

doi:10.1016/j.tcs.2010.09.006
fatcat:bwvs6vekt5a45lwze2xpxbs5ka
*of*an infinite avoiding word, in the case*of**sets**of*full words, in terms*of*two parameters: the*length*and the number*of*words in the*set*. ... The same algorithm also decides if the number*of*words*of**length*n avoiding a given finite*set**of*partial words grows polynomially or exponentially with n. ... When we consider*constant**length**sets**of*partial words, we implicitly require that neither the first or last position in any*of*the words be a hole. Proof. ...##
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On the Complexity of Deciding Avoidability of Sets of Partial Words
[chapter]

2009
*
Lecture Notes in Computer Science
*

We give a polynomial bound on the period

doi:10.1007/978-3-642-02737-6_9
fatcat:agy5mp2yyjgvhb6yenw6yguj7a
*of*an infinite avoiding word, in the case*of**sets**of*full words, in terms*of*two parameters: the*length*and the number*of*words in the*set*. ... The same algorithm also decides if the number*of*words*of**length*n avoiding a given finite*set**of*partial words grows polynomially or exponentially with n. ... When we consider*constant**length**sets**of*partial words, we implicitly require that neither the first or last position in any*of*the words be a hole. Proof. ...##
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Optimal Packings of 22 and 33 Unit Squares in a Square
[article]

2016
*
arXiv
*
pre-print

To achieve our results, we modify the well-known method

arXiv:1606.03746v1
fatcat:altddffd3fhcvbm2orj5gvhxju
*of**sets**of**unavoidable*points by replacing them with continuously varying families*of*such*sets*. ... Let s(n) be the side*length**of*the smallest square into which n non-overlapping unit squares can be packed. In 2010, the author showed that s(13)=4 and s(46)=7. ... By the same basic argument as applied to Figure 2 we can show that both the*set**of*red points and the*set**of*blue points form*unavoidable**sets*. ...##
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Page 789 of Mathematical Reviews Vol. 49, Issue 2
[page]

1975
*
Mathematical Reviews
*

]

*of*‘class-determin- ing measured*sets*’. ... The following notation is used: N denotes the*set**of*positive integers, X° denotes the*set**of*all infinite binary sequences, x(n) denotes the nth member*of*the binary sequence 2, x" denotes the initial ...##
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Complexity of Sequential Pattern Matching Algorithms
[chapter]

1998
*
Lecture Notes in Computer Science
*

Namely, we prove the exlstence

doi:10.1007/3-540-49543-6_16
fatcat:muwhtmqe5ng6fbjnhog2nvpmci
*of*"*unavoidable*positions" where the algorithm must stop to compare. ... Our results hold for any given pattern and text and for stationary ergodic pattern and text providing the*length**of*the pattern is order*of*magnitude smaller than the square root*of*the text*length*. ... The uniform bound [15] on the linearity*constant*, allows to define Ep(Et(cr,n))' when p ranges over a random (possibly infinite)*set**of*patterns. ...##
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Some characterizations of a class of unavoidable compact sets in the game of Banach and Mazur

1961
*
Pacific Journal of Mathematics
*

Steinhaus, an

doi:10.2140/pjm.1961.11.945
fatcat:4zbappmlmbdajonglhg43xwqtm
*unavoidable*perfect*set**of*measure 0 with the game-*constant*k -1 was constructed. ... Finally (p 3 ) A*set*S is said to have the property (p 3 ) if it is*unavoidable*(for the game*constant*k). ... PUBLISHED BY PACIFIC JOURNAL*OF*MATHEMATICS, A NON-PROFIT CORPORATION The Supporting Institutions listed above contribute to the cost*of*publication*of*this Journal, but they are not owners or publishers ...##
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Isosystolic inequalities for optical hypersurfaces
[article]

2016
*
arXiv
*
pre-print

This result is deduced from an interesting dual version

arXiv:1308.5522v2
fatcat:zrbyxoqiczajfhx7ieqsq6zgc4
*of*Minkowski's lattice-point theorem: if the origin is the unique integer point in the interior*of*a planar convex body, the area*of*its dual body ... We explore a natural generalization*of*systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry*of*numbers. ... The*set**of**unavoidable*convex bodies in R n whose volume is bounded above by some*constant*c > 0 is compact modulo unimodular transformations. ...##
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Strict Bounds for Pattern Avoidance
[chapter]

2013
*
Lecture Notes in Computer Science
*

Cassaigne conjectured in 1994 that any pattern with m distinct variables

doi:10.1007/978-3-642-38771-5_11
fatcat:cclt6tzmzzhm5a6yem6k5pzwy4
*of**length*at least 3(2 m−1 ) is avoidable over a binary alphabet, and any pattern with m distinct variables*of**length*at least 2 ... Building upon the work*of*Rampersad and the power series techniques*of*Bell and Goh, we obtain both*of*these suggested strict bounds. ... By Lemma 1, the number*of*words*of**length*n in S is at most [x n ]C(x), where By hypothesis, d i ⩾2 for 1⩽i⩽m. In order to use Theorem 1 on Σ , define and*set*the*constant*λ=k−0.08. ...##
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Strict bounds for pattern avoidance

2013
*
Theoretical Computer Science
*

Cassaigne conjectured in 1994 that any pattern with m distinct variables

doi:10.1016/j.tcs.2013.08.010
fatcat:qi6eqc6xs5cn5bie5ewur7ozx4
*of**length*at least 3(2 m−1 ) is avoidable over a binary alphabet, and any pattern with m distinct variables*of**length*at least 2 ... Building upon the work*of*Rampersad and the power series techniques*of*Bell and Goh, we obtain both*of*these suggested strict bounds. ... By Lemma 1, the number*of*words*of**length*n in S is at most [x n ]C(x), where By hypothesis, d i ⩾2 for 1⩽i⩽m. In order to use Theorem 1 on Σ , define and*set*the*constant*λ=k−0.08. ...##
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The Complexity of Unavoidable Word Patterns
[article]

2019
*
arXiv
*
pre-print

We present various complexity-related properties

arXiv:1901.07431v4
fatcat:xdj6hyrz35fbzkdenaatcs7lxi
*of**unavoidable*words. For words that are*unavoidable*, we provide an upper bound to the*lengths**of*words that avoid them. ... A natural subsequent question is how many*unavoidable*words there are. We show that the fraction*of*words that are*unavoidable*drops exponentially fast in the*length**of*the word. ... For n ∈ N we symbolize the*set**of*words*of**length*n over A by A n . ...##
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Page 6292 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

We restrict a

*set**of*distributions that a learning algorithm may use for its statistical queries to be a*set**of*product distributions with each bit being | with probability p, 1/2 or 1—p for a*constant*... (French summary) |*Unavoidable**sets*] Sém. Lothar. Combin. 47 (2001/02), Article B47e, 16 pp. (electronic). ...
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