## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 30, No. 4, 2004

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Contents

D.D. Anderson, Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419, U.S.A. (dan-anderson@uiowa.edu) and Tiberiu Dumitrescu, Facultatea de Matematică, Universitatea Bucureşti, 14 Academiei Str., Bucharest, RO 70109, Romania (tiberiu@al.math.unibuc.ro).
Half Condensed Domains, pp. 929-936.
ABSTRACT. An integral domain D is condensed (resp., strongly condensed) if for each pair of ideals I, J of D, IJ={ij ; i in I, j in J} (resp., IJ=iJ for some i in I or IJ =Ij for some j in J). In this paper we introduce and study the two related notions of a half condensed domain and a strongly half condensed domain. An integral domain D is half condensed if whenever a nonzero z is in IJ with I, J ideals of D, there exist I', J' (invertible) ideals of D such that I' is a subset of I, J' is a subset of J, and zD=I'J'. And D is strongly half condensed if whenever I, J are nonzero ideals of D, IJ=I1J for some invertible ideal I1 that is a subset of I or IJ=IJ1 for some invertible ideal J1 that is a subset of J.

John Harding and Guram Bezhanishvili, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-0001, USA (jharding@nmsu.edu), (gbezhani@nmsu.edu).
MacNeille Completions of Heyting Algebras, pp. 937-952.
ABSTRACT. In this note we provide a topological description of the MacNeille completion of a Heyting algebra similar to the description of the MacNeille completion of a Boolean algebra in terms of regular open sets of its Stone space. We also show that the only varieties of Heyting algebras that are closed under MacNeille completions are the trivial variety, the variety of all Boolean algebras, and the variety of all Heyting algebras.

Amir Khosravi, Faculty of Mathematical Sciences and Computer Engineering, University For Teacher Education, 599 Taleghani Ave., Tehran 15614, IRAN, and Behrooz Khosravi,  Dept. of Pure Math., Faculty of Math. and Computer Science, Amirkabir University of Technolog  (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, IRAN (khosravibbb@yahoo.com).
A New Characterization of Some Alternating and Symmetric Groups (II), pp. 953-967.
ABSTRACT. The order of every finite group G can be expressed as a product of coprime positive integers m1,...,mt such that the set of prime numbers divided mi is a connected component of the prime graph of G. The integers m1,...,mt are called the order components of G. Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184).
There exist some characterizations about alternating and symmetric groups. Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper, we suppose that p=2a xb+1>5 be a prime number, where a,b>0 are positive integers and x>3 is an odd prime number. Then by using the classification of finite simple groups, we proved that Ap, Ap+1, Ap+2, Sp, Sp+1, are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on An, where n=p, p+1 or p+2 are obtained. Also we generalize these conjectures for the groups Sn, where n=p, p+1.

Coffman, Adam, Indiana University - Purdue University Fort Wayne, Fort Wayne, IN 46805 (http://www.ipfw.edu/math/Coffman/).
Analytic Normal Form for CR Singular Surfaces in C3, pp. 969-996.
ABSTRACT. A real analytic surface inside complex 3-space with an isolated, non-degenerate complex tangent is shown to be holomorphically equivalent to a fixed real algebraic variety. The analyticity of the normalizing transformation is proved using a rapid convergence argument. Real surfaces in higher dimensions are also shown to have an algebraic normal form.

Hao Fang, Courant Institute of Mathematical Sciences, New York University, New York 10012, USA and Changyou Wang, Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA (cywang@ms.uky.edu).
On the Mean Curvature Flow for σk-Convex Hypersurfaces, pp. 997-1007.
ABSTRACT. We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is σk-convex.

Erdem, Sadettin, Middle East Technical University, 06531 Ankara, Turkey. (saerdem@fef.sdu.edu.tr ) or ( serdem@metu.edu.tr ).
J-Pseudo Harmonic Morphisms, Some Subclasses And Their Liftings To Tangent Bundles, pp. 1009-1038.
ABSTRACT. New subclasses of J- pseudo harmonic morphisms F of a (semi-) Rumanian manifold ( M, g) into a metric (para-) f-manifold (N,h,J ) are introduced, namely; nearly, quasi, semi homothetic harmonic maps. On the way, some characterizations of its tension field of F are given. Also liftings of J-pseudo harmonic morphisms F to the tangent bundles TM and TN, with various type of lifted metric (para-) f-structures are considered. Finally, some supporting examples are provided.

Garcia-Ferreira, S., Instituto de Matemáticas (UNAM), Apartado Postal 61-3, Xangari, 58089, Morelia, Michoacán, México (sgarcia@matmor.unam.mx), Sakai, S., Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan (sakaim01@kanagawa-u.ac.jp), and Sanchis, M., Departament de Matemátiques, Universitat Jaume I, Campus Riu Sec, 12071, Castelló, Spain (sanchis@mat.uji.es).
Free topological groups over ωμ-metrizable spaces , pp 1039-1053.
ABSTRACT. Let be an uncountable regular cardinal. For a Tychonoff space X, we let A(X) and F(X) be the free Abelian topological group and the free topological group over X, respectively. In this paper, we establish the next equivalences.
Theorem. Let X be a space. The following are equivalent.
1. (X,UX) is an -metrizable uniform space, where is the universal uniformity on X.
2. A(X) is topologically orderable and χ(A(X)) =ωμ .
3. The derived set is ωμ-compact and X is ωμ-metrizable.
Theorem. Let X be a non-discrete space. Then, the following are equivalent.
1. X is ωμ-compact and ωμ-metrizable.
2. (X,UX) is ωμ-metrizable and X is ωμ-compact.
3. F(X) is topologically orderable and χ(F(X)) =ωμ .
We also prove that an ωμ-metrizable uniform space (X,U) is a retract of its uniform free Abelian group A(X,U) and of its uniform free group F(X,U).

Naotsugu Chinen, University of Tsukuba, Ibraki 305-8571, Japan (naochin@math.tsukuba.ac.jp).
Sets of all ω-limit points for one-dimensional maps, pp. 1055-1068.
ABSTRACT. Let f be a continuous map from a graph G to itself and m the maximum of orders of all points of G. The main result of this paper is that a point c in G lies in the limit set of some point of G if and only if every open neighborhood of c contains at least (m + 1) points of some trajectory. This shows that every set of all limit points for every graph map satisfies the analogue to the Birkhoff theorem. But, the above does not holds for one-dimensional maps.

J.J. Charatonik, Instituto de Matematicas, UNAM, Cd. Universitaria, 04510 Mexico, D.F., Mexico (jjc@matem.unam.mx), W.J. Charatonik, Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65409-0020, U.S.A. (wjcharat@umr.edu) and J.R. Prajs, Department of Mathematics, Idaho State University, Pocatello, ID 83209, U.S.A. (prajs@isu.edu)
Atriodic absolute retracts for hereditarily unicoherent continua, pp. 1069-1087.
ABSTRACT. Let X be an absolute retract for the class of hereditarily unicoherent continua that contains no simple triod. In the paper we prove that (a) X is atriodic; (b) X is either an arc or an indecomposable continuum having only arcs as its proper subcontinua; (c) if X is either tree-like or circle-like, then it is arc-like.

Jonathan Hatch, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716 (hatch@math.udel.edu).
On a characterization of W-sets, pp. 1089-1101.
ABSTRACT. A proper subcontinuum H of a continuum X is said to be a W-set provided for each continuous surjective function f from a continuum Y onto X, there exists a subcontinuum C of Y that maps entirely onto H. Descriptions, definitions, and results concerning two new types of W-sets are given, as well as a new characterization of W-sets.

Louis Block, Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, (block@math.ufl.edu) and James Keesling, Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, (jek@math.ufl.edu).
Topological Entropy and Adding Machine Maps, pp. 1103-1113.
ABSTRACT. We prove two theorems which extend known theorems concerning periodic orbits and topological entropy in one-dimensional dynamics. Our first result may be described as follows. Given a sequence of prime numbers, we form the corresponding adding machine map (also called the odometer map). We then determine the infimum of the topological entropies of all continuous maps of the interval which contain a copy of the given adding machine map. Our second result deals with the following question. Suppose we are given a closed subset of the interval and a continuous map of this closed subset to itself. How do we extend the given map to a map of the entire interval which has the smallest possible entropy?

On removing one point from a compact space, pp. 1115-1126.
ABSTRACT. If B is a compact space such that after removing one point x it is still Lindelof, then any power of B satisfies that after removing one point (namely the point all whose coordinates are x) it is still star-Lindelof. If after removing one point B is still compact, then any power of B, after removing one point is still discretely star-Linedlof. In particular, this gives new examples of Tychonoff, discretely star-Lindelof spaces with unlimited extent.

R. Lowen and S. Verwulgen, Department of Mathematics, University of Antwerp, Antwerp 2020, Belgium (rlow@ruca.ua.ac.be), (vwulgen@ruca.ua.ac.be).
Approach Vector Spaces, pp. 1127-1142.
ABSTRACT. In this paper we determine what properties an approach structure has to fulfil for it to concord well with a vector space structure. Not surprisingly these conditions are more subtle than those for a topology. That the conditions we impose are the right ones follows mainly from the good categorical relationship among the different categories which play an important role in this setting, namely topological vector spaces, completely regular spaces, metrizable vector spaces and of course approach vector spaces.

Boulabiar, Karim , IPEST, University of Carthage, BP 51, 2070-La Marsa, Tunisia 42101 (karim.boulabiar@ipest.rnu.tn).
Order Bounded Separating Linear Maps on Φ-Algebras, pp.1143-1155.
ABSTRACT. A Φ-algebra is an Archimedean lattice ordered algebra with a weak order unit. Let be A and B be Φ -algebras and let T be a separating linear map from A into B, that is, T is a linear map such that T(f)T(g) = 0 in B whenever fg = 0 in A. It is proven by an order theoretical and purely algebraic method that there exist a 'weight' element w in B and a positive algebra homomorphism S from A into the maximal ring of quotients Q(B) of B such that T(f) = wS(f) holds for all f in A. Both real and complex cases are considered. This result generalizes the following theorem proved by W. Arendt in his paper [Spectral properties of Lamperti operators, Indiana Univ. J. Math., 32 (1983), 199-215]. Let C(X) and C(Y) be the Φ-algebras of all scalar-valued continuous functions on compact Hausdorff topological spaces X and Y, respectively. Then for every separating linear map T from C(X) into C(Y) there exist a 'weight' function w in C(Y) and a function h from Y into X (continuous on the cozero set of w) such that T(f)(y) = w(y)f(h(y)) holds for all f in C(X) and y in Y.

B.E. Forrest and L.W. Marcoux, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 (beforres@math.uwaterloo.ca) , (LWMarcoux@math.uwaterloo.ca).
Second Order Cohomology of Triangular Banach Algebras, pp. 1157-1176.
ABSTRACT. Explicit calculations of the various cohomology groups of a Banach algebra I are often very difficult to obtain. In this paper, we will present an elementary method to describe the second cohomology group H2(I ,I) for a class of algebras called triangular Banach algebras. The techniques are then illustrated through a number of examples.

Zhai Fahui, Institute of Mathematics, Institute of Qingdao Chemical Technology, Qingdao 266042 , P.R. China, (fahuiz@163.com).
The Closures of (u+k)-Orbits of Class Essentially Normal Operators, pp. 1177-1194.
ABSTRACT. Let A , B be simply connected analytic cauchy domains, B be a subset of the closure of A , u and v be measures on the boundary of A and B* which are assumed to be equivalent to arc length measures, respectively. M(A , u) and M(B*, v) are the multiplication operators on Hardy spaces of functions on the simply connected cauchy domain A and the simply connected cauchy domain B* , respectively. In this paper, we describe the closure of (u+k)-oribt of direct sum of the finitely direct sum M(A , u) with the finitely direct sum M(B*, v)*, furthermore, if A=D={z: |z|<1}, we also describe the closure of (u+k)-oribt of finitely direct sum of a class essentially normal operator models.

Leo Livshits, Department of Mathematics and Computer Science, Colby College, Waterville, ME 04901 (llivshi@colby.edu), Sing-Cheong Ong, Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859 (ong1s@cmich.edu ) and Sheng-Wang Wang Department of Mathematics, Nanjing Audit Institute, Nanjing 210029, China (wang2598@nju.edu.cn).
Schur Algebras Over Function Algebras, pp. 1195-1217.
ABSTRACT. The authors generalize results of L. Livshits, S.-C. Ong and S. W. Wang, Banach Space Duality in Absolute Schur Algebras, Integral Equations and Operator Theory. Vol. 41 (2001) 343-359.

Hem Raj Joshi, Department of Math and CS, Xavier University Cincinnati, OH 45207-4441 (joshi@xavier.edu) and Suzanne Lenhart, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300 (lenhart@math.utk.edu).
Solving a Parabolic Identification Problem by Optimal Control Methods, pp. 1219-1242.
ABSTRACT. An unknown coefficient of the interaction term of a parabolic system with a Neumann boundary condition in a multi-dimensional bounded domain is identified. The solution of the system represents the concentrations of prey and predator populations. Given partial (perhaps noisy) observations of a true solution in a subdomain, we seek to ``identify" the coefficient of the interaction term using an optimal control technique, involving Tikhonov's regularization. The existence and uniqueness of the optimal control approximating the desired coefficient are obtained, an optimality system is derived, the identification problem is discussed and an example illustrating how to find a solution numerically is presented.