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CUT POINTS IN ABCOHESIVE, APOSYNDETIC, AND SEMI-LOCALLY CONNECTED SPACES
DAVID A. JOHN and SHING S. SO Received 17 September 2001
In 1941, F. B. Jones introduced aposyndesis, which generalizes the concept of semi-local connectedness defined earlier by G. T. Whyburn (1942), in the study of continuum the- ory. Using Jones’s idea, D. A. John (1993) defined abcohesiveness as a generalization of aposyndesis and studied theA-sets in abcohesive spaces. In this paper, some properties of abcohesive spaces are studied and a number of results by B. Lehman (1976) and Whyburn (1942, 1968) are generalized; sufficient conditions for the existence of two nodal sets are established as well.
2000 Mathematics Subject Classification: 54D05.
1. Aposyndetic and abcohesive spaces
Definition1.1. A spaceMisaposyndetic at a point p with respect to a point qif and only if there exists a closed connected setHsuch thatp∈int(H)andH⊆M−{q}. The spaceMisaposyndetic at a point pif and only if it is aposyndetic atpwith respect toqfor eachqinM−{p}. The space isaposyndeticif and only if it is aposyndetic at pfor eachpinM.
A spaceM isabcohesive at a point p with respect to a pointqif and only if there exists an open connected setU such thatp∈U and U⊆M− {q}. The spaceM is abcohesive at a pointpif and only if it is abcohesive atpwith respect toqfor each qinM− {p}. The space isabcohesiveif and only if it is abcohesive atpfor eachp inM.
John [2] showed that aposyndesis implies abcohesiveness and hence abcohesivness is also a generalization of semi-local connectedness. Besides the properties discussed in [2], the next three theorems show some additional properties of abcohesive spaces.
All spaces in this paper are assumed to be topological.
Theorem1.2. (a)Every quasicomponent of an abcohesive space is open and is also a component of the space.
(b)If the setsHandKare abcohesive atp, thenH∪Kis abcohesive atp.
(c)IfMis a Hausdorff space,Sis an open subset ofMsuch thatSis abcohesive, then Sis abcohesive at every pointp∈S with respect to every pointq∈S−S.
Proof. (a) LetQbe a quasicomponent of the abcohesive spaceM. If M is con- nected, thenMis the only component of the space soQ=M. IfM is disconnected, letCbe a component inMsuch thatC∩Q≠∅. SinceC≠M, letp∈M−C. ThenCis
a component ofM−{p}and henceCis open by [2, Theorem 1]. SinceCis both open and closed,Q⊆C. But,Cis a subset of a quasicomponent ofM, thereforeC=Q.
(b) For eachq∈H∪K− {p},q∈H− {p}orq∈K− {p}so there exist open and connected setsU andV such thatq∈U⊂H−{p}orq∈V⊂K−{p}. ThusH∪Kis abcohesive.
(c) Letp∈Sandq∈S−S. SinceMis Hausdroff, there exist disjoint open setsUp
andVq such thatp∈Upandq∈Vq. Sinceq∈S−S, there existsq∗∈S such that q∗∈Vq. Since S is open and abcohesive, there exists an open connected set U of M such thatp∈U,U⊂S−q∗, andq∉U becauseq∈S−S. Therefore,p∈U and U⊂S−{q,q∗} ⊂S−{q}.
The next theorem is a characterization of abcohesive spaces.
Theorem1.3. A space is abcohesive if and only if each of its quasicomponents is open and abcohesive.
Proof. LetQbe a quasicomponent of the abcohesive spaceM. ByTheorem 1.2(a), it suffices to show thatQis abcohesive. For eachp∈Qandq∈Q−{p}, there exists an open connected setUsuch thatp∈U andU⊆M−{q}becauseMis abcohesive.
Sincep∈U and U is connected, byTheorem 1.2(a)U⊂Qand henceU is an open connected set inQ−{q}. Therefore,Qis abcohesive.
Conversely, assume that every quasicomponent ofM is open and abcohesive. For eachp∈M andq∈M− {p}, letQbe the quasicomponent ofM containingp and assume thatM≠Q. Letq∉Q. Sincep∈Q,Q⊂M−{q}, andQis open and connected by Theorem 1.2(a), M is abcohesive at p with respect to q. Let q ∈Q. SinceQ is abcohesive there exists an open connected subset U of Qcontaining p such that U⊂Q− {q} ⊂M− {q}. SinceQis open inM and U is open inQ, U is open inM. Thus,Uis an open connected set inM containingpsuch thatU⊂M− {q}soM is abcohesive atp with respect toqfor eachq∈Q− {p}andp∈Q. Therefore,M is abcohesive.
Definition1.4. LetMbe a connected space. A pointpis acut pointofMif and only ifM−{p}is disconnected. A pointpis anend pointofMif and only if each open set containingp contains an open set containingp whose boundary is degenerate.
Two pointsa,bin a spaceMareconjugateinMif no point ofMseparateaandb. Definition1.5. AnA-setAof a spaceMis a closed subset ofMsuch thatM−A is the union of a collection of open sets each bounded by a single point ofA.
Definition1.6. Leta and b be points of a spaceM. ThenC(a,b) denotes the intersection of all A-sets of M, which contain botha andb, and the set C(a,b)is called thecyclic chaininMfromatob.
Theorem1.7. Ifaandbare two distinct conjugate points of an abcohesive con- nected spaceM andC(a,b)≠∅, then C(a,b)= {x∈M:x is conjugate to botha andb}.
Proof. Letx be conjugate to botha and b and assume a∉ Afor some A-set A of M containing a and b. Let K be the component of M−A containing x. By
[2, Theorem 12], K is open, ∂K= {p}for some p in A, and K is a component of M−{p}. Now,M−{p} =K∪[(M−{p})−K]whereKand(M−{p})−Kare separated withx∈Kand{a,b}∩[(M−{p})−K]≠∅. Thus,xis not conjugate toaorb, which is a contradiction.
Letx∈Afor allA-setsAofMcontainingaandb. Supposexis not conjugate to a. Then there existsp∈Msuch thatM−{p} =Mx∪MawhereMxandMaseparated withx∈Mxanda∈Ma. LetK∗be the component ofM−{p}containinga. Ifb≠p and b∈Mx, then it contradicts the fact thata andb are distinct conjugate points ofM. Ifb=p or b∈Ma, thenK∗∪ {p}is anA-set ofM containing bothaand b according to [2, Theorem 14], sox∈K∗∪{p}, which is a contradiction.
The proof of the next theorem is similar toTheorem 1.3and hence is omitted.
Theorem1.8. Every quasicomponent of an aposyndetic space is aposyndetic.
Theorem1.9. IfMis a Hausdorff aposyndetic continuum andAis anA-set ofM, thenAis an aposyndetic continuum.
Proof. The fact thatA is a continuum follows from [2, Theorem 15]. For each p∈Aandq∈A−{p}, sinceM is aposyndetic, there exists a closed connected setH such thatp∈HandH⊂M−{q}. By [2, Theorem 15],H∩Ais closed and connected and int(H)∩Ais open inAcontainingpso the result follows.
Definition 1.10. Asimple chain is a finite collection {U1,...,Un}of point sets such thatUi∩Uj≠∅if and only if|i−j| ≤1.
Definition1.11. For any two pointsaandbof a connected spaceM, letE(a,b)= {x∈M:xseparatesafrombinM}. TheintervalabofM, denoted byIab, is the set E(a,b)∪{a,b}.
It is known that every interval in a connected semi-locally connected space is closed.
The following theorem shows that the result holds in a connected aposyndetic space.
Theorem1.12. Every interval of a connected aposyndetic Hausdorff space is closed.
Proof. LetIab be an interval of a connected aposyndetic Hausdorff spaceM. As- sumepis a limit point ofIab such thatp∉Iab. LetC be the component inM− {p}
containinga. Sincep∉Iab,bmust be inC. That is,{a,b} ⊂C. Suppose there exists anx∈Iab such thatx∉C. LetM− {x} =A∪B, whereAandBare separated with a∈Aand b∈B. Since{a,b} ⊂C⊆M− {x} =A∪B,C⊆A orC⊆B. This implies {a,b} ⊂Aor {a,b} ⊂B, which yields a contradiction. Thus ifx∈Iab, thenx∈C. For each x∈C, there exists a closed connected setHx such thatx ∈int(Hx)and p∉Hxso{int(Hx):x∈C}is an open cover ofC. By [1, Theorem 3.4], there exists a simple chain{int(Hx1),int(Hx2),...,int(Hxn)}fromatob. Suppose for somex∈Iab, x∉∪ni=1int(Hxi). LetM− {x} =A∪B, whereAandBare separated witha∈Aand b∈B. Since∪ni=1Hxiis connected,∪ni=1Hxi⊆Aor∪ni=1Hxi⊆B. This implies{a,b} ⊂A or{a,b} ⊂B, which is a contradiction. HenceIab⊆ ∪ni=1Hxi andp∉∪ni=1Hxi. The fact that∪ni=1Hxiis closed impliespis not a limit point ofIab. This contradicts the original assumption, henceIabis closed.
Theorem1.13. Leta,b, andpbe three distinct points inMsuch thatp∉Iab. IfM is Hausdorff, connected, and aposyndetic at each point inM− {p}, then there exists a closed connected subsetNofMwith{a,b} ⊂N⊂M−{p}.
Proof. Sincep∉Iab, there exists a component C ofM− {p}containing {a,b}.
SinceMis aposyndetic at each point inM−{p}, for eachx∈C, there exists a closed connected setKxsuch thatx∈int(Kx)⊂Kx⊂M− {p}soC⊂ ∪x∈Cint(Kx). By [1, Theorem 3.4], there is a simple chain{Kx1,...,Kxn}fromatobandp∈ ∪ni=1Kxi. The result follows by lettingN= ∪ni=1Kxi.
Corollary1.14. IfM is Hausdorff, connected, and aposyndetic at each point in M−{p}, wherepis a noncut point ofM, then for each pair of pointsa,binM−{p}, there exists a closed connected subsetNofMwith{a,b} ⊂N⊂M−{p}.
Definition1.15. A subsetE of a spaceM is anE0-set ofM if and only ifE is nondegenerate, connected, has no cut point of itself, and is maximal with respect to these properties.
Definition1.16. LetR be a relation on the nondegenerate connected spaceM defined by xRy if and only if no point ofM separates x fromy inM. Thus R is reflexive and symmetric onM. For each pointxofMwhich is neither a cut point nor an end pointM, then the set of all pointsyofMsuch thatxRyis called asimple link ofM.
Definition1.17. A spaceM isparaseparableif and only ifM does not contain uncountably many disjoint open sets.
The next lemma is known and can be found in [4].
Lemma1.18. IfYis an uncountable set of cut points of the paraseparable connected setM, then some two points ofY are separated inMby a third point ofY.
Lemma 1.19. EveryE0-set of a paraseparable connected set M contains at most countably many cut points ofM.
Proof. Suppose that there exists anE0-setEofMsuch that the setY, which is the set of points inEthat are also cut points ofM, is uncountable. ByLemma 1.18, some two pointsa,bofY are separated inMby a third pointxofY. LetM−{x} =A∪B, whereAandBare separated witha∈Aandb∈B. ThenE⊂A∪{x}orE⊂B∪{x}, which implies{a,b} ⊂Aor{a,b} ⊂B. This contradicts the statement thata,b are separated byxinM.
Theorem1.20. LetMbe a connected, locally compact, paraseparable, and aposyn- detic space. IfLis a nondegenerate subset ofM, thenLis a simple link ofMif and only ifLis anE0-set ofM.
Proof. LetLpbe a simple link containingpinM. Suppose thatLphas a cut point x. ThenLp− {x} =A∪B, whereAandB are separated inLpwitha∈A, b∈B for somea,binLp. Sincexcannot separateaandbinM, byTheorem 1.13, there exists a continuumNsuch that{a,b} ⊂N⊂M−{x}. Without loss of generality, assume that
Nis irreducible betweenaandb. SinceN⊂M− {x}andLpcontains all irreducible continua containingaand b by [6, Chapter IV, Theorem 1.3],N⊂Lp− {x}implies N⊂AorN⊂B, which contradicts{a,b} ⊂N. Therefore,Lphas no cut point.
Suppose that there is a connected setCsuch thatLp⊂C andChas no cut point.
Letx∈C−Lp. Then there existsy∈Msuch thatM−{y} =Mx∪Mp, whereMxand Mpare separated withx∈Mxandp∈Mp. SinceChas no cut point of itself,y∉C. Then{x,p} ⊂C⊂M−{y}impliesx∉Mxorp∉Mp, which leads to a contradiction.
Therefore,Lpis anE0-set.
Conversely, letEbe anE0-set. SinceEis closed andMis locally compact,Eis locally compact and hence is uncountable because no locally compact connected set is the union of a nondegenerate countable collection of disjoint compact sets. SinceEis an E0-set andEcontains at most countably many cut points ofM, there existsp∈Esuch thatpis neither a cut point nor an end point ofMbyLemma 1.19. LetLpbe the simple link containingp. SupposeE ⊆Lp. Letq∈E−Lp. Then there existsx∈M such that M−{x} =Mp∪Mq, whereMpandMqare separated withp∈Mpandq∈Mq. Ifx∉E, thenE⊂MporE⊂Mq. This implies{p,q} ⊂Mpor{p,q} ⊂Mq, which contradicts the fact thatpandqare separated byxinM. Ifx∈E, thenE−{x} =(E∩Mp)∪(E∩Mq). This implies thatxis a cut point ofE, which contradicts the fact thatEis anE0-set.
Therefore,E⊆Lp. Since it has been proved that every simple link is anE0-set in a locally compact, paraseparable, aposyndetic space,Lp=E.
2. Semi-locally connected spaces
Definition2.1. A spaceMissemi-locally connectedat a pointpofMif and only if each open set containingp contains an open setV containingp such thatM−V has at most a finite number of components. The spaceMissemi-locally connectedif and only if it is semi-locally connected at each pointpofM.
Theorem 2.2. Every quasicomponent of a semi-locally connected space is semi- locally connected.
Proof. LetQbe a quasicomponent of a semi-locally connected spaceMandp∈Q. SinceQis an open subset ofMcontainingpandMis semi-locally connected, there is an open setV inMcontainingpsuch thatp∈VandM−V= ∪ni=1Ki, whereKiis a component ofM−V. NowQ−V=Q∩(M−V )=Q∩(∪ni=1Ki), letF= {j:Kj∩Q≠
∅, j=1,2,...,n}. Then for eachj ∈F,Kj⊂QbecauseQis a component ofMby Theorem 1.2(a), andKjis a component ofQ−V. Therefore,Q−V= ∪j∈FKi, which implies thatQis semi-locally connected.
Definition2.3. Acyclic elementof a connected spaceMis a subset ofMwhich either consists of a single cut point or end point ofMor is anE0-set ofM.
Theorem2.4. (a)IfMis a paraseparable semi-locally connected continuum andA is a closed set inM, thenAis anA-set inMif and only if for each cyclic elementEof Msuch thatA∩Eis nondegenerate,E⊂A.
(b)IfMis a paraseparable semi-locally connected continuum andAis a nondegen- erate subcontinuum inM, thenAis anA-set inMif and only ifAis the union of cyclic elements ofM.
(c)IfMis a paraseparable semi-locally connected continuum, then every cyclic ele- ment ofMis anA-set ofM.
Proof. (a) LetAbe anA-set ofMand letEbe a cyclic element ofMsuch thatA∩E is nondegenerate. Letpandqbe distinct points ofA∩E. Assume there exists a point xinE−A. LetKbe the component ofM−Acontainingxand let∂K= {y}. SinceEis connected and it intersects bothKandM−K,y∈E. But thenyis a cut point ofE. Now, eitherporqis inA∩E−{y}. Therefore,E⊂A.
Conversely, suppose that, for each cyclic elementE ofM such thatA∩E is non- degenerate,E⊂A. Then the result follows from [6, Chapter IV, Theorem 3.3] and [2, Theorem 12].
(b) The result follows from [6, Chapter IV, Theorem 3.3] and [2, Theorem 12].
(c) LetEbe a cyclic element ofM and letp∈E. Then either{p} =E, or there is a cyclic element, namelyE, ofM such thatp∈E andE⊂E. By part (a),E is anA-set ofM.
Theorem2.5. IfMis a paraseparable semi-locally connected continuum, then the intersection of a collection ofA-sets ofMis itself anA-set ofM.
Proof. Let Ᏻ be a collection of A-sets of M, and let A= ∩Ᏻ. If A≠∅ and is degenerate, thenAis anA-set ofM. Leta∈A, and for eachxinA− {a}, letNxbe the irreducible subcontinuum ofMfromatox. By [2, Theorem 18], for everyG∈Ᏻ, Nx⊂G, so thatNx⊂A. HenceAis a continuum. Suppose p∈Aand{p}≠A. Ifp is a cut point or an end point ofM, then{p}is a degenerate cyclic elements of M and{p} ⊂A. Ifp is a non-cut point and a non-end point ofM, thenp belongs to a nondegenerate cyclic elementEofM. ByTheorem 2.4(a),E⊂Gfor eachG∈Ᏻ. Then E⊂A, and hence, again byTheorem 2.4(a),Ais anA-set.
Definition2.6. IfᏳis a covering of a spaceM, then a subsetEofMis said to be ofdiameterless thanᏳif some element ofᏳcontainsE. LetᏲbe a family of subsets of a spaceM. ThenᏲis called anull familyif and only if for every open coverᏳofM, all but a finite number of members ofᏲhave diameter less thanᏳ.
The following theorem established by Simone [5] is useful in proving the result: in a semi-locally connected space, the set of all components of the complement of an A-set is a null family.
Theorem2.7. LetᏳbe a family of subsets of the compact spaceM. ThenᏳis a null family if and only if for any two disjoint closed setsAandB, at most a finite number of members ofᏳintersect bothAandB.
Theorem2.8. IfM is a semi-locally connected continuum andAis anA-set, then the set of all components ofM−Ais a null family.
Proof. LetHandKbe two disjoint closed subsets ofM. Let{Ci:i∈I}be the set of components ofM−Athat intersect bothHandK. AssumeIis infinite. For eachi∈I, let∂Ci= {pi}. If{pi:i∈I}is finite, then for some pointp∈A,pi=pfor infinitely manyCi. If{pi:i∈I}is infinite, then letp be a limit point of{pi:i∈I}. SinceA is closed,p∈A. One of the setsH orK, sayK, does not containp. LetU=M−K.
ThenK⊂M−U and(M−U)∩Ci≠∅for infinitely manyi∈I. But this contradicts the fact thatMis semi-locally connected. HenceIis finite. ByTheorem 2.7, the set of all components ofM−Ais a null family.
3. Concerning nodal sets
Definition3.1. A spaceMis said to bestrongly connectedif and only if for each two pointsaandbinM, there exists a continuumLinMsuch thatLcontainsaand b. A subsetNof a connected spaceM is called anodal set ofM if and only ifNis closed and∂Nis degenerate.
Note thatNis anA-set and is connected.
Definition3.2. A spaceMisconnected im kleinenat the pointpofMif and only if each open setUcontainingpcontains an open setV containingpsuch that each point ofVbelongs to a connected set containingpand lying inU.
Theorem3.3. IfMis a nondegenerate paraseparable connected space andUis an open subset ofMsuch thatUis locally compact and each nondegenerate continuum in Ucontains uncountably many cut points ofM, then there exist two pointsaandb in Mthat are separated inMby uncountably many points ofM.
Proof. Letp∈U. There exists an open setV such thatp∈V, V⊂U, andV is compact. AssumeMis not connected im kleinen at p. There exists a continuum of convergenceKinV such thatp∈K. ThenKcontains uncountably many cut points ofM. ByLemma 1.18, some two points in Kare separated inM by a third point of K. But this is impossible, sinceK is a continuum of convergence inM. HenceM is connected im kleinen atp. ThenUis connected im kleinen atp, and therefore,Uis locally connected.
LetWbe an open connected set such thatW⊂U. SinceW is locally compact,Wis strongly connected. Letaandbbe two points inW,Lbe an irreducible continuum in W fromatob, and letT be the set of all points ofLthat separatesafrombinM. AssumeT is countable. LetS=(L− {a,b})−T. ThenS contains uncountably many cut points ofM. Hence, byLemma 1.18, some two points ofS are separated inM by a third pointxofS. By the irreducibility ofL,xseparatesafrombinM. Butx∉T, and this is a contradiction. HenceT is uncountable.
Theorem 3.4. IfM is a connected space, a andb are points of M, and Lis an irreducible continuum inMfromatob, thenL−{a,b}contains no end point ofM.
Proof. LetLbe an irreducible continuum inM fromatob. AssumeLcontains an end pointpofM. LetU be an open set containing psuch thataandb are not inU. There exists an open setV such thatp∈V,V⊂U, and∂V is degenerate. Let
∂V= {q}. ThenL∩Uis separated fromL∩(M−V ), and soqis a cut point ofL. Now L∩[(M−V )∪{q}]is a proper subcontinuum ofLfromatob. This is a contradiction.
HenceLcontains no end point ofM.
Theorem3.5. IfMis a nondegenerate paraseparable connected space andUis an open subset ofMsuch thatU is locally compact andU contains at most a countable
number of non-cut points ofMthat are non-end points ofM, then there exist two points aandbinMthat are separated inMby uncountably many points ofM.
Proof. LetU be as described in the hypothesis. LetK be a nondegenerate con- tinuum inU. ThenKis uncountable, andKcontains at most a countable number of non-cut points ofM that are non-end points of M. Letx and y be distinct points ofK, and letLbe an irreducible subcontinuum ofKfromx toy. ByTheorem 3.4, L−{x,y}contains no end point ofM, and soLcontains uncountably many cut points ofM. Then byTheorem 3.3, there exist two pointsaandbinMthat are separated in Mby uncountably many points ofM.
The following lemma is known and can be found in [4].
Lemma3.6. Ifaandbare points of the paraseparable connected spaceMandH is an uncountable subset ofIab, then there exists a countable subsetKofHsuch that ifpis a point ofH−K, thenM−{p}is the union of two separated connected sets one containingaand the other containingb.
Theorem3.7. IfMis a nondegenerate paraseparable connected space andUis an open subset ofMsuch thatUis locally compact and each nondegenerate continuum of Ucontains uncountably many cut points ofM, then there exist two disjoint nodal sets HandKofMsuch thatM−HandM−Kare connected andHandKhave nonempty interiors.
Proof. ByTheorem 3.4andLemma 3.6, there exists an uncountable subsetTofU such that each point ofT separatesMinto two nonempty connected sets. Letq∈T, and letM−{q} =A∪B, whereAandBare two nonempty separated connected sets.
Let p∈T∩A. ThenM− {p} =E∪F, whereE and F are two nonempty separated connected sets. SinceE∪{p}is a closed connected subset ofA∪B,E∪{p} ⊂A. Also B∪{q}is a closed connected subset ofM. It is easy to see thatE∪{p}andB∪{q}are disjoint. Now letH=E∪{p}andK=B∪{q}. ThenM−H=F andM−K=A, andF andAare both connected. SinceEandBare open,HandKare closed. HenceHand Khave nonempty interior and degenerate boundaries.
Lehman [3] stated the theorem: ifMis a connected space,pis a cut point ofMand M− {p} =H∪K, thenH∪ {p}andK∪ {p}are nodal sets. Whyburn [6] established a similar theorem. These theorems follow immediately from the definition of nodal sets. The following theorem provides sufficient conditions for the existence of two nodal sets in nondegenerate locally compact paraseparable connected spaces; how- ever, these nodal sets are disjoint and do not separate the space.
Theorem3.8. IfM is a nondegenerate locally compact paraseparable connected space, then there exist two disjoint nodal setsHandKofMsuch thatM−HandM−K are connected.
Proof. If each point ofMis a cut point ofM, then the result follows fromTheorem 3.7. Ifpandqare distinct non-cut points ofM, then letH= {p}andK= {q}. IfMhas exactly one non-cut pointx, thenU=M− {x}. NowU is locally compact, and each
nondegenerate subcontinuum ofUcontains uncountably many cut points ofM. The results then follows fromTheorem 3.7.
References
[1] J. G. Hocking and G. S. Young,Topology, Addison-Wesley, Massachusetts, 1961.
[2] D. A. John,A-sets and abcohesive spaces, Missouri J. Math. Sci.5(1993), no. 2, 63–67.
[3] B. Lehman,Cyclic element theory in connected and locally connected Hausdorff spaces, Canad. J. Math.28(1976), no. 5, 1032–1050.
[4] R. L. Moore,Foundations of Point Set Theory, American Mathematical Society Colloquium Publications, vol. 13, American Mathematical Society, Rhode Island, 1962.
[5] J. N. Simone,Hereditarily locally connected continua and the Hahn-Mazurkiewicz problem, Ph.D. thesis, University of Missouri, Kansas City, 1976.
[6] G. T. Whyburn,Analytic Topology, American Mathematical Society Colloquium Publica- tions, vol. 28, American Mathematical Society, Rhode Island, 1942.
David A. John: Missouri Western State College, Saint Joseph, MO64507, USA E-mail address:[email protected]
Shing S. So: Central Missouri State University, Warrensburg, MO64093, USA E-mail address:[email protected]
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according to the following timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
José Roberto Castilho Piqueira,
Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;
[email protected]
Elbert E. Neher Macau,
Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected]
Celso Grebogi,Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]
Hindawi Publishing Corporation http://www.hindawi.com
