WEAK CONFLUENCE AND MAPPINGS TO ONE-DIMENSIONAL POLYHEDRA

Transcription

1 Voume 8, 1983 Pages WEAK CONFLUENCE AND MAPPINGS TO ONE-DIMENSIONAL POLYHEDRA by Pamea D. Roberso Topoogy Proceedgs Web: Ma: Topoogy Proceedgs Departmet of Mathematcs & Statstcs Aubur Uversty, Aabama 36849, USA E-ma: ISSN: COPYRIGHT c by Topoogy Proceedgs. A rghts reserved.

2 TOPOLOGY PROCEEDINGS Voume WEAK CONFLUENCE AND MAPPINGS TO ONE-DIMENSIONAL POLYHEDRA Pamea D. Roberso 1. Itroducto Throughout ths paper the term mappg w mea a cotuous fucto ad a cotuum w be a compact, coected metrc space. Suppose X s a cotuum, K s a subcotuum of X, ad f s mappg of a cotuum oto X. The statemet that f s weaky cofuet wth respect to K meas some compoet of f-(k) s throw by f otok. The statemet that f s weaky cofuet meas f s weaky cofuet wth respect to each subcotuum of X. Ay mappg of a cotuum oto a tree s weaky cofuet wth respect to each arc whch does ot cota a jucto pot ts teror. May peope, such as Read [7, Lemma p. 236], Igram [3, Lemma 1], ad Marsh [5, Lemma 4.7] have gve a proof of some verso of ths, usg the fact that the teror of such a arc separates the tree. Feurerbacher [2, Lemma 9] showed that f K s a arc a crce S the ay mappg of a cotuum oto S must be weaky cofuet wth respect to K or S-K. I Theorem 4 of ths paper we show that f K,,K s a coecto of subcotua of a oe-dmesoa poyhedro X whose terors are mutuay excusve ad cota o jucto pots, the the foowg are equvaet. (1) Ay mappg of a cotuum oto X s weaky cofuet wth respect to oe of K,,K, ad

3 196 Roberso (2) The uo of the terors of K,,K separate X. I Theorem 5 we gve codtos o the poyhedro whch sure the separato (2) above. We the use verse mt represetatos of oe-dmesoa poyhedra to gve codtos uder whch ay mappg of a cotuum oto a oe~dmesoa poyhedro X must be weaky cofuet wth respect to oe of a gve coecto of subcotua of X. The theorems ths paper ca be used to show that certa oe-dmesoa cotua are Cass(W), where Cass(W) s the cass of cotua whch are mages of weaky cofuet mappgs oy. We gve a exampe of how these theorems may be used. 2. Weak Cofuece ad Separato ocoe-dmesoa Poyhedra paper. I ths secto we estabsh the ma theorems of the Theorem 1. Suppose X s a oe-dmesoa coected poyhedro ad K,K,,K are mutuay excusve o 2 degeerate subcotua of X, o oe of whch cotas a jucto pot or a edpot of x. equvaet: The the foowg are (1) If f s a mappg of a cotuum oto X the f s weaky cofuet wth respect to oe of K,K,,K, 2 ad (2) X - U~=K s ot coected. Proof (1).. (2): Suppose X - U=K s coected. Let A,A,,A be a mutuay excusve co 2 X such that A ~ It K for = 1,2,,. The

4 TOPOLOGY PROCEEDINGS Voume X - U=A s coected ad we deote by M the cotuum. X - u=a We defi)e a mappg f of M oto X whch s ot weaky cofuet wth respect to ay K - For = 1,2,,, K - A s the uo of two mutuay excusve arcs u of whch has oe edpot whch s a edpot of K oe edpot whch s a edpot of A. ad S each ad We defe fu ad fs so that fu s a homeomorphsm whch maps u oto 1. u. U A. ad fs. s a homeomorphsm whch maps S oto S U ad so that the edpot of K beogg to u. s 1. A' 1. a fxed pot of f u ad the edpot of K. beogg to 1. S s a fxed pot of f Is. We defe f (X U=K to ) be the detty mappg o X - U=K S. For = 1,2,, f-(k. ) has two compoets, u ad Nether f(u ) or f(s) s K, hece f s ot weaky cofuet wth respect to K. (2) ~ (1): Suppose X - (U=K ) s ot coected. Let f be a mappg of a cotuum M oto x. Case 1. X - K s ot coected. Let A be a arc X cotag o jucto pot or edpot of X such that K ~ It A. The X - A s ot coected ad has oy two compoets, C ad C 2 Let a E C A ad a E C A Let g be the mappg of X oto A defed by f x E C g(x) f x E C 2 = {:: f x E A. The composto gof s a mappg of M oto the arc A ad so by [7, Lemma p. 236] gof s weaky cofuet. Thus,

5 198 Roberso there s a subcotuum H of M such that gof(h) = K. f(h) s a cotuum X whch s throw by g oto K, ga -1 s a homeomorphsm, ad g (K ) = K the f(h) = K,. Therefore, f s weaky cofuet wth respect to K. Case 2. X - K s coected. Let m be a postve teger ess tha such that X - U~ K. s coected ad 1= 1 Sce X - U=K m+ 1S. ot coecte. d Let A b e t ua 11,A 2,,A m + mu y excusve arcs X, o oe of whch cotas a jucto pot or a edpot of X, such that K ~ It A for m m+ =,,m+. The X - U=A s coected ad X - U=A s ot coected. Sce X - U~:A = (X - U~=A) - A +, m m+ the X - U=A has oy two compoets, C ad C 2. Let a E A C ad a E A C Let g be the mappg of X oto A defed by g(x) = 1: 1 f x E A f x E C a f x E C 2 2 ad for = 2,,m+, we defe ga. to be a homeomorphsm 1 whch throws A oto A 1 such a way that g(k ) = K 1, g(a C ) = {a, ad g(a C 2 ) = {a 2 }. The composto gof s a mappg of M oto the arc A, ad so by [7, Lemma p. 236] gof s weaky cofuet. there s a subcotuum H of M such that gof(h) = K 1. Thus, Sce f(h) s a cotuum X whch s throw by 9 oto K, g-(k ) U~:K' ad ga s a homeomorphsm for =,,m+, the f(h) s oe of K,,K m +. Therefore, f s weaky cofuet wth respect to oe of K,,K +. m

6 TOPOLOGY PROCEEDINGS Voume The ext theorem gves codtos whch sure the separato (2) of Theorem 1. To each metrc space X there correspods a o-egatve teger b (X) (see [4, p. 409]). If X s a poyhedro, b(x) s the oe-dmesoa Bett umber of X. Theorem 2. Suppose X s a oe-dmesoa coected poyhedro ad s o-egatve teger such that b (X), ad K,K,---,K + are mutuay excusve subcotua of 2 X, o oe of whch cotas a jucto pot or a edpot of x. The X - U~:tK s ot coected. Proof Suppose X - U +K' = 1S coec t e d Let A,A,---,A + 2 be mutuay excusve arcs X, o oe of whch cotas a jucto pot or a edpot of X, such that K ~ It A' for = 1,2,---,+. coected. Let a be a edpot of A ad g be the mappg of X to X defed by I: g(x) = otherwse Sce g s a mootoe mappg, t foows from [4, Theorem 4, p. 433] that b(x) ~ b(g(x». But g[x] has oy + smpe cosed curves ad oe jucto pot; hece, b(g(x» = + 1. Ths yeds a cotradcto. The ext theorem foows from Theorem 1 ad 2. Theorem 3. Suppose X s a oe-dmesoa coected poyhedro, s a o-egatve teger such that b(x) =,

7 200 Roberso ad K,K 2,,K + are mutuay excusve o-degeerate subcotua of X~ or a edpot of x. o oe of whch cotas a jucto pot If f s a mappg of a cotuum oto X the f s weaky cofuet wth respect to oe of I Theorems 4, 5, ad 6 we reax the codtos regardg jucto pots mposed o the coectos of subcotua the hypotheses of Theorems 1, 2 ad 3. Theorem 4. Suppose X s a oe-dmesoa poyhedro~ K,,K are o-degeerate subcotua of X whose terors are mutuay excusve, ad o oe of K,K, K cotas 2 a jucto pot of X ts teror. The the foowg are equvaet. (1) If f s a mappg, of a cotuum oto X the f s weaky cofuet wth respect to oe of K,K2,,K~ ad (2) X - u=it K s ot coected. Proof (1) => (2): Suppose X - u=it K s ot coected. Let f be a mappg of a cotuum M oto x. For each = 1,2,,, et A,A, be a sequece of 2 arcs such that A. c It K A. c A.+ ad m A. = K.. J -, J - J, j~oo J The for each postve teger j, A.,A.,,A. are mutuay J J J excusve subcotua of X, o oe of whch cotas a jucto pot or a edpot of X.. d h. td Th 1S ot coecte, t e X - u. Ao 1S ot coec e. e, 1= J by Theorem 1, f s weaky cofuet wth respect to oe of 1 2 A.,A.,,A.. J J J

8 TOPOLOGY PROCEEDINGS Voume There exsts a postve teger such that f s weaky cofuet wth respect to ftey may of A~A2'. Thus, there s a sequece L,L, of subcotua of M 2 such that f(l ),f(l ), s a subsequece of A,A,. 2 2 We choose a subsequece L,L, of L,L 2,. whch m m 2 coverges to a subcotuum L of M. The f(l) m f(l ). m. J-+OO J m A~ K. Therefore, f s weaky cofuet wth respect j-+oo J to K.. 1 (2) =- (1): Suppose that X - U=It K s coected. Let A,A,,A be arcs X such that A C It K for 2 = 1,2,,. The X - U=A s coected, ad so, by Theorem 1, there exsts a cotuum M ad a mappg f of M oto X such that f s ot weaky cofuet wth respect to A' for each 1,2,,. Sce, for each, K - A s ot coected, t foows from Theorem 1 that f s ot weaky cofuet wth respect to K, for each = 1,2,.,. Theorem 5. Suppose X s a oe-dmesoa coected poyhedro ad s a o-egatve teger such that b(x) = ad K,K,,K + are subcotua of X whose 2 terors are mutuay excusve, ad o oe of K,K,,K 2 + cotas a jucto pot of X ts teror. The X - U~~~It K s ot coected. + Proof. Suppose X - U=It K s coected. Let A,A,,A + be subcotua of X such that A ~ It K, 2 for = 1,2,,+. The A,A,,A + are mutuay 2 excusve subcotua of X, o oe of whch cotas a jucto pot or a edpot of X, ad X - u~~~ s coected. Ths cotradcts Theorem 2.

9 202 Roberso The ext theorem foows from Theorems 4 ad 5. Theorem 6. Suppose that X s a oe-dmesoa coected poyhedro, s a o-egatve teger such that b(x) =, ad K I,K,,K + are o-degeerate subcotua 2 1 of X whose terors are mutuay excusve, ad o oe of K I,K,,K + cotas a jucto pot of X ts teror. 2 1 If f s a mappg of a cotuum oto X the f s weaky cofuet wth respect to oe of K I,K 2,,K +. Theorem 5 shows that a oe-dmesoa coected poyhedro X, ay coecto of at east b(x) + I subcotua of X whch satsfy certa codtos must separate X. The foowg theorem shows that t s ecessary to requre ths may subcotua to assure separato. Theorem 7. Suppose X s a oe-dmesoa coected poyhedro ad s a postve teger such that b(x) =. pot of X, such that X - curve. Proof. The there exst mutuay excusve subcotua K,K,,K I 2 of X, o oe of whch cotas a jucto pot or a ed U=K s coected. Sce b(x) > I the X cotas a smpe cosed Let K I be a arc ths smpe cosed curve whch cotas o jucto pot of X. The X - K s coected, I so by the Euer-Pocare formua [6, Theorem 9, p. 32] (Oe ca see ths by otg that X - K I has oe more I-smpex ad two more a-smpexes tha X. ) We defe, ductvey, arcs K,,K X such that 2 for j- j = 2,, K s a smpe cosed curve X - U=IK, j

10 TOPOLOGY PROCEEDINGS Voume j- K j cotas o jucto pot of X, ad X - U=K s co ected. By the Euer-Pocare formua, b (X - U~ K.) 1= 1 Therefore, X - U=K s coected. I the ext theorem, we show that the codtos regardg jucto pots mposed o the coecto of subcotua Theorem 4 may ot be weakeed. Theorem 8. Suppose X s a oe-dmesoa coected poyhedro ad K,K,,K are mutuay excusve proper 2 subcotua of X such that each of K,K,,K.cotas a 2 jucto pot of X ts teror. The there exsts a cotuum M ad a mappg f of M oto X such that f s ot weaky cofuet wth respect to K for each = 1,2,,. Proof. We show there s a cotuum M ad a mappg f of M oto X whch s ot weaky cofuet wth respect to K. There s a pot x X - U=K ad a arc a = [x,a] such that a E K [x,a) (U~ K.) = ~, ad [x,a) cotas, 1= 1 't' o jucto pot of X. Let J be a jucto pot of X It K 1 ad et S = [a,j] be a arc K 1 jog a ad J. Case 1. (U=K ) = <P There s a arc [t,j] such that [t,j) ad [t,j) cotas o jucto pot of X. Let [k,j] be a arc S such that [k,j) cotas o jucto pot of X. Let M be the uo of the foowg three subsets of X x [0,1]:

11 204 Roberso [X - (k, J)] x {O}, (ex U S u [t, J]) x {I}, ad {X,t} x [0,1]. Let f be the projecto mappg of the cotuum M oto X. to K. We show that f s ot weaky cofuet wth respect Suppose there s a subcotuum H of M such that f(h) = K. Sce J s the teror of K there s a pot y K such that y f ex U S U [k,j]. Now, -1-1 f (y) = {(y, 0) }, f [ (k, J)] = (k, J) x { 1 }, ad f [(k,j) x {I}] s oe to oe. Thus, H must cota the pot (y,o) ad a pot of (k,j) x {I}. But, ay subcotuum of M whch cotas (y,o) ad a pot of (k,j) x {I} must tersect oe of {t} x [0,1] ad {x} x ro,]. Therefore, the mage of such a cotuum uder f must cota a pot ot K, ad so f s ot weaky cofuet wth respect to K. Case 2. Case 1 does ot hod. The there exst two arcs [r,j] ad [s,j] such that [r,j) U [s,j) ~ K - Sad ether [r,j) or [s,j) cotas a jucto pot of X. We w resove ths case two parts. Frst, suppose that X - (r,j) s ot coected. The X - (r,j) has oy two compoets, oe cotag r ad the other cotag J, s, ad x. Let M be the uo of the foowg three subsets of X x [0,1]: [X (r,j)] x {a}, (ex U S u [r, J]) x {I}, ad {x,r} x [0,1]. Let f be the projecto mappg of M oto X.

12 TOPOLOGY PROCEEDINGS Voume We show that f s ot weaky cofuet wth respect to If H s ay subset of M such that K ~ f(h) K the H must cota the pot (s,o) ad a pot of (r,j) x {I}. But, ay cotuum M cotag two such pots must tersect {x} x [0,1], ad hece f(h) cotas pots ot K. Thus, f s ot weaky cofuet wth respect to K O the other had, suppose that X - (r,j) s coected. Let M be the uo of the foowg three subsets of X x [0,1]: [X - (r,j)] x {O}, (ex U S u [ r, J]) x { }, ad {x} x [0,1]. Let f be the projecto mappg of the cotuum Mato X. We show that f s ot weaky cofuet wth respect to K If H s ay subcotuum of M such that K ~ f(h). the H must cota (r,j) x {I} ad the pot (s,o). But, ay cotuum M cotag such pots must tersect {x} x [0,1] ad hece f(h) cotas pots ot K Thus, Ths co f s ot weaky cofuet wth respect to K. cudes Case 2. I each case, M was costructed by removg a arc from X ad budg a brdge over t X x [0,1]. I dog ths we were carefu to stay away from U=2K. Ths costructo ca be repeated for each of K,,K, resutg a co 2 tuum M' X x [0,1] such that the projecto mappg f' of M' oto X s ot weaky cofuet wth respect to K, for each 1,2,,.

13 206 Roberso Remapk. It s terestg to ote that wth M 1 so costructed, oe ca see that a arc ca be mapped oto M 1 such a way that the composto of ths mappg wth f s ot weaky cofuet wth respect to ay K. Thus, we may assume the cotuum M the statemet of Theorem 8 s a arc. 3. Iverse Lmts I ths secto we use verse mt represetatos of oe-dmesoa poyhedra to descrbe codtos uder whch ay mappg of a cotuum oto a oe-dmesoa poyhedro X must be weaky cofuet wth respect to oe member of a gve coecto of subcotua of X. These resuts ca be used to show that certa oe-dmesoa cotua are Cass(W). Suppose x,x 2, s a sequece of compact metrc spaces each havg dameter ess tha a fxed postve umber c, ad suppose f,f 2, s a sequece of mappgs such that f maps X + oto X for = 1,2,. The vepse Zmt of the verse mt sequece {X,f } s the subset of the product IT X. to whch (x,x, ) beogs f ad oy f >O 2 1. x for = 1,2,. We cosder IT X. metrzed >O 1. - d(x,y) IT 2 d. (x., y. ) >O J. where d deotes the metrc o X. For each = 1,2,, TI w deote the projecto mappg of the verse mt oto X.. 1.

14 TOPOLOGY PROCEEDINGS Voume The foowg emma was essetay proved by Read [7, Theorem 4] athough ot stated ths way. A proof s cuded here oy for the sake of competeess. Lemma 1. Suppose X s the verse mt of the verse mt sequece {X.,f.} wth each X. a cotuum~ 111 K s a subcotuum of X~ ad g s a mappg of a cotuum oto X. If TIog s weaky cofuet wth respect to TI(K) for ftey may tegers ~ the g s weaky cofuet wth respect to K. Proof. Let g be a mappg of a cotuum M oto X, 1, 2, 3, be a sequece of tegers, ad H 1,H 2, be a sequece of subcotua of M such that TI og(h.) = TI (K ) 1 for = 1,2,. We ca assume that the sequece H 1,H 2,H 3, coverges to a cotuum H M. We show that K c g(h). Suppose p K ad E > O. Let N be a postve teger such that f k > N the L 2- < E. > k Let k > N. Sce x K, TI (p) TI (K) = TI (H ). Let x k k k k be a pot of g(h ) such that TI (x) TI (p). The for k k k < k, TI(x) = TI(p). Thus, d(p,x) L 2 - d. (TI. (x), TI. (p) ) >O <, ad so d(p,g(h < for k > N. Hece, p E m g(h ) k» k-+-oo k g(h). Ths shows that K c g(h). We show g(h) c K. Suppose t g(k) ad E > O. Let N be a postve teger such that f k > N the L 2- < ~. > k Choose k > N such that g(h) c B(g(Hk)'~) ad et y be a pot of ~(Hk) such that d(t,y) < I. Sce y f(h k ) the

15 208 Roberso TI (y) E TI og(h ) = TI (H). There s a pot s H such k k k k that TI (y) = TI (s), ad so for < TI; (y) = TI; (s). k,... k k - E Thus, d (y, s) = : 2 d. (TI. (y), TI. (s» < '2' ad d (t, s) < >O J. J. J. d(t,y) + d(y,s) < '2 + 2 = E. E E Sce for each E > 0 there s a pot x H such that d(r,s) < E, the s E H = H. Ths shows th"at g (H) c K. We have show that g(h) K, thus g s weaky cofuet wth respect to K. I the foowg emma, d deotes the Hausdorff metrc. Lemma 2. Suppose X s a cotuum, K s a subcotuum of X, ad g s a mappg of a cotuum oto X. If fop each postve umbep E thepe s a subcotuum L of X such that g s weaky cofuet wth pespect to Lad d(k,l) < E the g s weaky cofuet wth pespect to K. Ppoof. The proof of ths emma s straghtforward. The ext two theorems foow easy from the emmas ad Theorems 4 ad 6 of secto 2. Theopem 9. Suppose X s the vepse mt of the vepse mt sequece {X,f } wth each X a oe-dmesoa coected poyhedpo, ad K1,,K ape o-degeepate sub cotua of X such that fop f~ey may tegeps, (1) the tepops of TIK,,TIK ape mutuay excusve, (2) o oe OfTIK,,TIK cotas a jucto pot of X; ts tepop, ad (3) X. - U. It(TI.K.) s ot co.. J. J= J. J. ected. If g s a mappg of a cotuum oto X the g s weaky cofuet wth pespect to oe of K,,K.

16 TOPOLOGY PROCEEDINGS Voume Theorem 10. Suppose X s the verse mt of the verse mt sequece {X.,f.} wth each X. a oe-dmesoa 111 coected poyhedro~ ad s a postve teger such that b(x ) ~ for each. Suppose aso that K,,K + are o-degeerate subcotua of X such that for ftey may tegers ~ (1) the terors of TIK,,TIK+ are mutuay excusve ad (2) o oe of TIK,,TIK+ cotas a jucto pot of x ts teror. If g s a mappg of a cotuum oto X the g s weaky cofuet wth respect A speca case of Theorem 9 was proved by Read [7# Theorem 4]. Theorems 9 ad 10 may be used to show that certa oe-dmesoa cotua are Cass(W). The foowg are cotua for whch Theorem 9 or 10 ca be used to show they are Cass(W): (1) the Cass(W) cotua defed by Waraszkewcz [9] (ot a of the cotua he descrbed are Cass(W», (2) the Case-Chamber cotuum [1], (3) Igram's cotua [3], ad (4) the cotuum defed by Sherg [8]. As a exampe, we w use Theorem 9 to show that the Case-Chamber cotuum s Cass(W). The Case-Chamber cotuum (see [1]) s a verse mt o fgure eghts usg oe bodg map. Let A ad B be two crces taget at a pot J. Assg a oretato to each of A ad B. Let f be a mappg whch throws A U B oto A U B as foows:

17 210 Roberso (1) A s throw oto A U B by fxg J, the wrappg aroud A the postve drecto, the B the postve drecto, ad the aroud each of A ad B the egatve drecto. (2) B s throw oto A U B by fxg J, the wrappg aroud A twce the postve drecto, the B twce the postve drecto, ad the aroud each of A ad B twce the egatve drecto. For each et X. = A U Bad f. = f. Let X be the 1 1 verse mt of the verse mt sequece {X,f }. ca sow that f K s a proper subcotuum of X the there exsts a postve teger such that (1) for each >, Oe J ~ TIK, or (2) for each >, TIK s a arc A havg J as a edpot. We w show that X s Cass(W). Let g be a mappg of a cotuum oto X ad et K be a proper subcotuum of X. We w show that for every postve umber E there s a subcotuum L of X such that g s weaky cofuet wth respect to Lad d(k,l) < E (where d deotes the Hausdorff metrc) We assume IT (X.,d.) metrzed by d(x,y) = >O E 2 d(ti.x,ti.y). Let E > 0 ad N be a postve teger >O 1 1 such that E 2- E There exsts a teger N > J >N dam(a U B) such that J s ot the teror of TIK. We ca choose mutuay excusve arcs a ad S A such that f(a) f(s) = TIK ad J s ot the teror of a or S. There exst subcotua L ad L of X such that 2

18 TOPOLOGY PROCEEDINGS Voume TI+(L ) = a, TI + (L 2 ) = S ad for each > +, TI(L ) ad TI (L ) are mutuay excusve arcs A, ether of 2 whch cotas J ts teror. The for >, X - [TI(L ) U TI (L 2 )] s ot coected. By Theorem 9, g s weaky cofuet wth respect to L or L 2 Sce TI (L ) = TI (L 2 ) = TI (K), the d(k,l ) < - L 2 (dam A U B) < E, ad < d(k,l ) < - L 2 (dam A U B) < E. 2 < Therefore, for each postve umber E there s a subcotuum L of X such that g s weaky cofuet wth respect to Lad d(k,l) < E. By Lemma 2, g s weaky cofuet wth respect to K. Hece, X s Cass(W). Refereces [1] J. H. Case ad R. E. Chamber, Charaaterzatos of tree-zke aotua, Pacfc J. Math. 10 (1960), [2] G. A. Feurerbacher, WeakZy ahaabze araze-zke aotua, Doctora Dssertato, Uversty of Housto, Housto, [3] W. T.O 'Igram, Coaerg atroda tree - Zke aotua, Fud. Math. 101 (1978), [4] K. Kuratowsk, TopoZogy II, Academc Press, New York, [5] M. M. Marsh, Fxed pot theorems for aerta tree-zke aotua, Doctora Dssertato, Uversty of Housto, Housto, [6] L. S. Potryag, Foudatos of aombatoraz topozogy, Grayock Press, Rochester, New York, [7] D. R. Read, CofZuet ad rezated ~appgs, Cooq. Math. 29 (1974), [8] D. D. Sherg, Coaerg the aoe=hyperspaae property, Ca. J. Math. (to appear).

19 212 Roberso [9] Z. Waraszkewcz, Sur u probzeme Math. 22 (1934), de M. H. Hah, Fud. Uversty of Housto Housto, Texas ad Stephe F. Aust State Uversty Nacogdoches, Texas 75962