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### Periodic sets of integers

Armando B. Matos
1994 Theoretical Computer Science
., Periodic sets of integers, Theoretical Computer Science 127 (1994) 287-312.  ...  Consider the following kinds of sets: _ the set of all possible distances between two vertices of a directed graph; _ any set of integers that is either finite or periodic for all n greater or equal to  ...  Type 2 Ultimately periodic sets of integers Type 3 Ultimately periodic sets of integers Theorem 3.1. Every ultimately periodic set is denoted by a A-sum.Proof.  ...

### Syntactic Complexity of Ultimately Periodic Sets of Integers [chapter]

Michel Rigo, Élise Vandomme
2011 Lecture Notes in Computer Science
We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b.  ...  We apply our results to some well studied problem: decide whether or not a brecognizable sets of integers is ultimately periodic.  ...  A set X of integers is said to be b-recognizable if the language rep b (X) ⊆ A * b is a regular language accepted by some DFA. A set X ⊆ N is periodic of period p if for all n ∈ N, n ∈ X ⇔ n + p ∈ X.  ...

### Accurate Computation of Periodic Regions' Centers in the General M-Set with Integer Index Number

Wang Xingyuan, He Yijie, Sun Yuanyuan
2010 Discrete Dynamics in Nature and Society
One method fits for the general M-sets with integer index number, the other fits for the general M-sets with negative integer index number.  ...  We primarily discuss the general M-sets with negative integer index, and analyze the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric  ...  interior, then comparatively analyze the relation of periodic regions' number in general M-sets with opposite integer index number.  ...

### Necessary and sufficient conditions for a sum-free set of positive integers to be ultimately periodic

Neil J. Calkin
2018
We shall denote the set of sum-free sets of positive integers by S.  ...  Introduction There is a natural bijection between the set of binary sequences and the set of sum-free sets of positive integers.  ...

### On almost periodic solutions of a class of differential equations

G. H. Meisters
1959 Proceedings of the American Mathematical Society
Considers From this last equation we conclude that the set of e-translation integers r of e(k) is identical to the set of e-translation integers of x(k).  ...  It is well-known that every almost periodic function possesses a relatively dense set of e-translation integers for each positive real number e.  ...  closure of the range of 4>(t), is a uniformly continuous family of almost periodic movements in C".  ...

### On Almost Periodic Solutions of a Class of Differential Equations

G. H. Meisters
1959 Proceedings of the American Mathematical Society
Considers From this last equation we conclude that the set of e-translation integers r of e(k) is identical to the set of e-translation integers of x(k).  ...  It is well-known that every almost periodic function possesses a relatively dense set of e-translation integers for each positive real number e.  ...  closure of the range of 4>(t), is a uniformly continuous family of almost periodic movements in C".  ...

### Linear forms and complementing sets of integers [article]

Melvyn B. Nathanson
2008 arXiv   pre-print
If this representation function is constant, then the set B is periodic and the period of B will be bounded in terms of the diameter of the finite set phi(a_1,...  ...  . + u_hx_h+vy be a linear form with nonzero integer coefficients u_1,..., u_h, v. Let A = (A_1,..., A_h) be an h-tuple of finite sets of integers and let B be an infinite set of integers.  ...  For example, if A is a finite set of integers and if B is an infinite set of integers such that the pair (A, B) is complementing, then B must be a periodic set, that is, a union of congruence classes modulo  ...

### Problems in additive number theory, II: Linear forms and complementing sets

Melvyn B. Nathanson
2009 Journal de Théorie des Nombres de Bordeaux
For example, if A is a finite set of integers and if B is an infinite set of integers such that the pair (A, B) is complementing, then B must be a periodic set, that is, a union of congruence classes modulo  ...  We shall prove that the set B is periodic, and obtain an upper bound for the period of B in terms of the diameter D ψ A of the finite set ψ(A).  ...

### Pattern Periodic Coloring of Distance Graphs

Xuding Zhu
1998 Journal of combinatorial theory. Series B (Print)
Let Z be the set of all integers.  ...  The problem of determining the chromatic numbers of distance graphs on the integers has received much recent attention [1, 2, 4, 5, 11 18, 20]. The cases that |D| =1 or 2 are easy [1, 17] .  ...  We say a coloring c of the set Z of integers is a pattern periodic coloring if there is an integer p, and a permutation f of the set of colors such that for every x # Z, we have c(x)= f (c(x& p)).  ...

### The 3n+l-Problem and Holomorphic Dynamics

Simon Letherman, Dierk Schleicher, Reg Wood
1999 Experimental Mathematics
A periodic point z is a point for which f n (z) = z; the period of z is the least positive integer n for which this equation holds.  ...  We see that every integer sits either in the basin of attraction of a superattracting periodic orbit of integers, or it is in a wandering Fatou component.  ...

### Page 116 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 10, Issue 1 [page]

1959 American Mathematical Society. Proceedings of the American Mathematical Society
MEISTERS [February From this last equation we conclude that the set of ¢-translation integers rt of e(k) is identical to the set of €-translation integers of x(k).  ...  It is well-known that every almost periodic func- tion possesses a relatively dense set of €-translation integers for each positive real number e. (Cf. [1, Paragraph 4°, p. 54]).  ...

### Mappings of the Interval with Finitely Many Periodic Points Have Zero Entropy

Louis Block
1977 Proceedings of the American Mathematical Society
It is shown that if / is a continuous map of a closed interval into itself, and / has finitely many periodic points, then the topological entropy of / is zero.  ...  Suppose that f has finitely many periodic points, and all periodic points of f are fixed points off. Let F denote the union of the set of fixed points of f and the set of endpoints of I.  ...  Let n be the product of the periods of all periodic points of /. Then all periodic points of/" are fixed points of/". By Theorem B, ß(/") is the set of fixed points of /".  ...

### Mappings of the interval with finitely many periodic points have zero entropy

Louis Block
1977 Proceedings of the American Mathematical Society
It is shown that if / is a continuous map of a closed interval into itself, and / has finitely many periodic points, then the topological entropy of / is zero.  ...  Suppose that f has finitely many periodic points, and all periodic points of f are fixed points off. Let F denote the union of the set of fixed points of f and the set of endpoints of I.  ...  Let n be the product of the periods of all periodic points of /. Then all periodic points of/" are fixed points of/". By Theorem B, ß(/") is the set of fixed points of /".  ...

### Periodicity of complementing multisets [article]

Zeljka Ljujic
2010 arXiv   pre-print
Let A be a finite multiset of integers. If B be a multiset such that A and B are t-complementing multisets of integers, then B is periodic.  ...  Then B is periodic with period k≤ (diam(A)+1)^1/3+ϵ.  ...  If B is a set such that A and B are t-complementing sets of integers with respect to ρ, then B is periodic with period log k ≤ (diam(ψ(A)) + 1) 1 3 +ε .  ...

### Asymptotically Periodic Behaviour in the Dynamics of Chaotic Mappings

H. E. Nusse
1987 SIAM Journal on Applied Mathematics
Moreover almost every point is asymptotically periodic with period p, for some positive integer p.  ...  We will prove that, for a chaotic mapping f belonging to a suitable class of C + functions, the set Ao(f) has Lebesgue measure zero, with A(f) a nonempty set consisting of points whose orbits do not converge  ...  We set U(k)for the component of LJ{Dj;O<-j<-N(k)+N(f)-I} such that YN(k) Bd (U(k)). Let X(k) be a boundary point of UN(k) and XS(k) YN(k).  ...
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