Tamás Bérczes a , Gábor Guta b , Gábor Kusper c Wolfgang Schreiner b , János Sztrik a
4. Conclusions and future work
We carried out an initial analysis of hierarchical decompositions of transformation semigroups using the holonomy algorithm. We showed that when working with the components’ state sets we have to deal with covers that are not images of the covered set. We also found a sharp upper bound for the width of the decomposition.
However, other properties of the holonomy decomposition, including its height, still need further investigation.
References
[1] Arbib, M. A.,editor, Algebraic Theory of Machines, Languages, and Semigroups, Academic Press, 1968.
[2] Dilworth, R. P., A decomposition theorem for partially ordered sets, Annals of Mathematics, 51 (1950) 161–166.
[3] Dömösi, P., Nehaniv, C. L., Algebraic Theory of Finite Automata Networks: An Introduction, volume 11. SIAM Series on Discrete Mathematics and Applications, 2005.
[4] Egri-Nagy, A., Nehaniv, C. L., On straight words and minimal permutators in finite transformation semigroups. LNCS Lecture Notes in Computer Science, 2010.
Proceedings of the 15th International Conference on Implementation and Application of Automata CIAA, in press.
[5] Egri-Nagy, A., Nehaniv, C. L., SgpDec– software package for hierarchical co-ordinatization of groups and semigroups, implemented in theGAP computer algebra system, Version 0.5.25+, 2010. http://sgpdec.sf.net.
[6] Eilenberg, S., Automata, Languages and Machines, volume B, Academic Press, 1976.
[7] Ganyushkin, O., Mazorchuk, V., Classical Transformation Semigroups,Algebra and Applications, Springer, 2009.
[8] Ginzburg, A., Algebraic Theory of Automata, Academic Press, 1968.
[9] Holcombe, W. M. L. Algebraic Automata Theory, Cambridge University Press, 1982.
[10] Rhodes, J., Steinberg, B., The q-theory of Finite Semigroups, Springer, 2008.
[11] Zeiger, H. P.,Cascade synthesis of finite state machines,Information and Control, 10 (1967) 419–433, plus erratum.
[12] Zeiger, H. P., Yet another proof of the cascade decomposition theorem for finite automata, Math. Systems Theory, 1 (1967) 225–228, plus erratum.
84 A. Egri-Nagy, C. L. Nehaniv Attila Egri-Nagy
Eszterházy Károly College
Institute of Mathematics and Informatics Department of Computing Science Eger, Leányka út 4, Hungary e-mail: attila@egri-nagy.hu Chrystopher L. Nehaniv
Royal Society Wolfson BioComputation Research Lab
Centre for Computer Science & Informatics Research, University of Hertfordshire Hatfield, Hertfordshire AL10 9AB, United Kingdom
e-mail: C.L.Nehaniv@herts.ac.uk
Annales Mathematicae et Informaticae 37(2010) pp. 85–93
http://ami.ektf.hu
On the best estimations for dispersions of special ratio block sequences ∗
Ferdinánd Filip
a, Kálmán Liptai
bFerenc Mátyás
b, János T. Tóth
aaDepartment of Mathematics, J. Selye University
bInstitute of Mathematics and Informatics, Eszterházy Károly College Submitted 4 October 2010; Accepted 24 November 2010
Dedicated to professor Béla Pelle on his 80th birthday Abstract
Properties of dispersion of block sequences were investigated by J. T. Tóth, L. Mišík, F. Filip [20]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence.
Keywords:dispersion, block sequence,(R)-density.
MSC:Primary 11B05.
1. Introduction
In this part we recall some basic definitions. Denote by N and R+ the set of all positive integers and positive real numbers, respectively. For X ⊂Nlet X(n) =
#{x∈X;x6n}. In the whole paper we will assume thatX is infinite. Denote by R(X) ={xy;x∈X, y ∈X} theratio set of X and say that a setX is(R)-dense if R(X)is (topologically) dense in the setR+. Let us notice that the concept of (R)-density was defined and first studied in papers [17] and [18].
Now letX ={x1, x2, . . .}wherexn< xn+1are positive integers. The sequence x1
x1
,x1
x2
,x2
x2
,x1
x3
,x2
x3
,x3
x3
, . . . ,x1
xn
,x2
xn
, . . . ,xn
xn
, . . . (1.1)
∗Supported by grants APVV SK-HU-0009-08 and VEGA Grant no. 1/0753/10.
85
86 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth of finite sequences derived fromX is calledratio block sequenceof the setX. Thus the block sequence is formed by blocksX1, X2, . . . , Xn, . . .where
This kind of block sequences were studied in papers, [1] , [3] , [4] , [16] and [20]. Also other kinds of block sequences were studied by several authors, see [2], [6], [8], [12]
and [19]. Let Y = (yn)be an increasing sequence of positive integers. A sequence of blocks of type
was invetigated in [11] which extends a result of [5]. Authors obtained a complete theory for the uniform distribution of the related block sequence(Yn).
For everyn∈Nlet the maximum distance between two consecutive terms in the n-th block.
In this paper we will consider the characteristics (see [20]) D(X) = lim inf
n→∞ D(Xn),
called the dispersionof the block sequence (1.1) derived fromX, and its relations to the previously mentioned asymptotic density of the original set X.
At the end of this section, let us mention the concept of a dispersion of a general sequence of numbers in the intervalh0,1i. Let(xn)∞n=0be a sequence inh0,1i. For can be found for example in [10] where it is also proved that
lim sup
N→∞
N dN > 1 log 4
holds for every one-to-one infinite sequencexn∈ h0,1i. Also notice that the density of the whole sequence (xn)∞n=0 is equivalent to lim
N→∞dN = 0. Also notice that the analogy of this property for the concept of dispersion of block sequences defined in the present paper does not hold.
Much more on these and related topics can be found in monograph [13].
On the best estimations for dispersions of special ratio block sequences 87
2. Results
When calculating the valueD(X), the following theorems are often useful (See [20], Theorem 1, Corollary 1, respectively).
(A1) Let
(A2) Let X be identical to the form of X in (A1). Suppose that there exists a positive integern0 such that for all integersn > n0
cn+1−dn6cn+2−dn+1.
The basic properties of the dispersionD(X)and the relations between dispersion and(R)-density are investigated in the paper [TMF]. The next theorem states the upper bound for dispersionsD(X)of(R)-dense sets where16a= lim
n→∞ n∈Nand suppose that the limit lim
n→∞
The following theorem shows that in the third case (ifa>2), that the dispersion D(X)can be any number in the interval
88 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth (A4) Leta>1 be a real number andk be an arbitrary natural number. Then for everyα∈ h0,aka2k−1ithere exists an(R)-dense set
In this paper we prove that in the second case ifa∈D
. The following lemma will be useful.
Lemma 2.1. Let the set
M(X) ={n∈N:cn+1−dn= max{ci+1−di:i= 1,2, . . . , n}}=
Proof. Letn∈Nbe an arbitrary integer such thatn>m1. Then there is unique k∈Nwithmk6n < mk+1. >From the definition of the setM(X)we obtain
k=1is a subsequence of the sequence max The last two inequalities imply
D(X) = lim inf
On the best estimations for dispersions of special ratio block sequences 89 Theorem 2.2. Let a ∈ 1+√5
2 ,2
be an arbitrary real number. Then for every α∈ h0,a12ithere is an (R)-dense set increasing on the interval ha,∞). Moreover
f(a) = a−1
Obviously for everyn∈N a < bn,1
First we prove thatD(X) =α. We have the following inequalities:
cn+1,1−bn+1,2>bbn+1,2−bn+1,2>(b−1)bn+1,2>(b−1)a2an+1,1>
>(a−1)a2an+1,1>aan+1,1> an+1,1> an+1,1−dn,n
90 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth
From the above inequalities we have for a sufficiently large n ∈ N the following inequalities:
1 =an,2−bn,1< an,1−dn−1,n−1< cn,1−bn,2< cn,2−dn,1< . . .
· · ·< cn,n−dn,n−1< an+1,1−dn,n. (2.1) Now we use Lemma 2.1. From (2.1) one can see that it is sufficient to study the quotients:
On the best estimations for dispersions of special ratio block sequences 91
and it is sufficient to prove that the ratio set of the setX is dense on intervals 1
which is obviously a subset of the ratio set of X. The largest difference between consecutive terms of (2.2) is bn,22 . Then
an,1+ 1 hence the ratio set of X is dense in the interval
1 a2,1E
.
Letl∈Nbe arbitrary. We prove that the ratio set ofX is dense in the interval 1
which is obviously a subset of the ratio set of X. The largest difference between consecutive terms of (2.3) is 6a 2
n,1+1. On the other hand,
92 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth
References
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[20] Tóth, J. T., Mišík, L., Filip, F., On some properties of dispersion of block sequences of positive integers,Math. Slovaca, 54 (2004), 453–464.
On the best estimations for dispersions of special ratio block sequences 93 Ferdinánd Filip,János T. Tóth
Department of Mathematics J. Selye University
Bratislavská cesta 3322 945 01 Komárno Slovakia
e-mail: filip.ferdinand@selyeuni.sk toth.janos@selyeuni.sk Kálmán Liptai,Ferenc Mátyás Institute of Mathematics and Informatics Eszterházy Károly College
H-3300 Eger Leányka út 4.
Hungary
e-mail: liptaik@ektf.hu matyas@ektf.hu
Annales Mathematicae et Informaticae 37(2010) pp. 95–100
http://ami.ektf.hu