Electronic Edition Vol. 24, No. 3, 1998

Editors: G. Auchmuty (Houston), H. Brezis (Paris), S. S. Chern (Berkeley), J. Damon (Chapel Hill), K. Davidson (Waterloo), L. C. Evans (Berkeley), R. M. Hardt (Rice), J. A. Johnson (Houston), A. Lelek (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), G. Pisier (College Station and Paris), R. Scott (Houston), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)


Yoshio Agaoka, Department of Mathematics, Faculty of Integrated Arts & Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan (agaoka@mis.hiroshima-u.ac.jp).
A new example of higher order almost flat affine connections on the three-dimensional sphere, pp.387-396
An error has been corrected in Vol. 24, No. 4, 1998.
ABSTRACT. We give new examples of torsion free affine connections on the three-dimensional sphere and Brieskorn manifolds with almost vanishing curvature, by considering a class of left invariant affine connections on Lie groups. These examples indicate a striking difference between "Riemannian" and "affine" category in considering the concept "almost flatness".

J. Llibre, Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Spain (jllibre@manwe.math.uab.es), J. Paraños, Departamento de Anàlisi Matemàtica, Facultade de Matemàticas, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain, and J.A. Rodríguez, Departamento de Matemàticas, Facultad de Ciencias, Calvo Sotelo s/n, Universidad de Oviedo, 33007 Oviedo, Spain.
Periods for Transversal Maps on Compact Manifolds with a Given Homology, pp. 397-407.
ABSTRACT.Let M be a compact  differentiable manifold such that its rational homology groups are either Q or 0.  A differentiable map f: from M into itself  is called transversal if for all m in N the graph of the m-th iterate of f  intersects transversally the diagonal of M x M at each point (x,x) such that x is a fixed point of the m-iterate of f.. We study the set of periods of f by using the Lefschetz numbers for periodic points.

Rukimbira, Philippe, Florida International University, Miami, FL 33199 (rukim@fiu.edu).
A characterization of flat contact metric Geometry, pp.409-414.
ABSTRACT. Flat contact metrics exist only in dimension 3. Among the six isometry classes of flat closed 3-manifolds, one consists of manifolds with trivial first de Rham cohomology. We prove that this particular classe admits no flat contact metric by showing that a flat closed contact 3-manifold carries a nonsingular parallel vector field and thus has nontrivial first de Rham cohomology.

Lei Fu, Institute of Mathematics, Nankai University, Tianjin 300071, China (leifu@sun.nankai.edu.cn).
An Analogue of Bernstein's Theorem, pp. 415-419.
ABSTRACT. We prove the following analogue of Bernstein's theorem: Let f(x,y) be a smooth function defined on the whole plane. Then the graph of the gradient of f(x,y) is a minimal surface if and only if f(x,y) is harmonic or a quadratic polynomial.

Bejancu, Aurel, Technical University of Iasi, 6600 Iasi, Romania, (relu@math.tuiasi.ro), Hernández Encinas, Luis, University of Salamanca, Paseo de Canalejas 169, 37008 Salamanca, Spain, (encinas@gugu.usal.es), and Muñoz Masqué, Jaime, CSIC, Serrano 144, 28006 Madrid, Spain, (jaime@iec.csic.es).
Invariant Differential Forms on the First Jet Prolongation of the Cotangent Bundle, pp.421-442.
ABSTRACT. The structure of the differential forms on J1(T*M) which are invariant under the natural representacion of the gauge algebra of the trivial principal bundle, MxU(1), and the structure of the horizontal forms on the J1(T*M) which are invariant under the Lie algebra of all infinitesimal automorphisms of MxU(1) are determined.

D. Buhagiar, Department of Mathematics, Shimane University, Matsue 690-8504, Japan (buhagiar@math.okayama-u.ac.jp) and T. Miwa, Department of Mathematics, Shimane University, Matsue 690-8504, Japan (miwa@riko.shimane-u.ac.jp).
On Superparacompact and Lindelöf GO-Spaces, pp. 443-457.
ABSTRACT. In this paper we study some compact/paracompact type properties, namely weak superparacompactness, superparacompactness and Lindelöfness. Particular attention is given to GO-spaces. It is proved that a GO-space X is weakly superparacompact if and only if every gap is a W-gap and every pseudogap is a W-pseudogap. A characterization of Lindelöf GO-spaces involving C-(pseudo)gaps is given. We also show that there is a 1--1 correspondence between superparacompact (resp. Lindelöf) GO-d-extensions and preuniversal ODF (resp. prelindelöf) GO-uniformities. Finally we give several examples corresponding to the above results.

C.E.M. Pearce, Applied Mathematics Department, University of Adelaide, Adelaide S.A. 5005, Australia (cpearce@maths.adelaide.edu.au), and J. Pecaric, V. Simic, Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 41000 Zagreb, Croatia.
Weighted Generalized Logarithmic Means, pp. 459-465.
ABSTRACT. An integral representation of Neuman is extended and used to suggest a multidimensional weighted generalized logarithmic mean. Some inequalities are established for such means. A number of known results appear as special cases.

Chun-Lan Jiang, Department of Mathematics, Jilin University, Chang Chun, Jilin, People's Republic of China (cljiang@ns1.hebut.edu.cr), and Pei Yuan Wu, Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China (pywu@cc.nctu.edu.tw).
Sums of strongly irreducible operators, pp. 467-481.
ABSTRACT. In 1969, Radjavi proved that every (bounded linear) operator on a complex separable Hilbert space is the sum of two irreducible operators.
In this paper, we consider the more refined problem whether every operator is even the sum of two strongly irreducible operators. We are able to show that for certain classes of operators this does have an affirmative answer. These classes include those of finite-dimensional operators, triangular operators, multicyclic operators and compact operators. In general, we can only show that every operator is the sum of three strongly irreducible operators.

R.L. Moore, and T.T. Trent, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350 (rmoore@gp.as.ua.edu).
Solving Operator Equations in Nest Algebras, pp. 483-488.
ABSTRACT. Let X and Y be operators on Hilbert space, and let L be a nest of projections on the space. We consider the problem of finding an operator A in Alg(L) such that A is Hilbert-Schmidt and such that AX = Y. A necessary and sufficient condition involving X, Y, and the projections in the lattice is found. We also indicate how the statements of the results can be modified so that the main theorem is true for any commutative subspace lattice L.

Robbins, D. A., Department of Mathematics, Trinity College, Hartford, CT 06106 (david.robbins@mail.trincoll.edu).
BSE Banach modules and bundles of Banach spaces, pp. 489-505.
ABSTRACT. In a recent paper [J. Funct. Anal. 125 (1994), 67-89], S.-E. Takahasi defined the notion of a BSE Banach module over a commutative Banach algebra A with bounded approximate identity. We show that the multiplier algebra M(X) of X can be represented as a space of sections in a bundle of Banach spaces, and we use bundle techniques to obtain shorter versions of various of Takahasi's results on C*-algebra modules and to answer several questions which he raised.

Ronald G. Douglas Texas A & M University at College-Station, TX (rgd@tamu.edu) and Rongwei Yang Texas A & M University at College-Station, TX and Dept. of Mathematics, SUNY at Stonybrook, NY (rwyang@math.tamu.edu)
Quotient hardy Modules, pp. 507-517.
ABSTRACT. Suppose H2(Dn) is the Hardy space over the unit polydisk Dn, and [h] is the closed submodule generated by a bounded holomorphic function h in Hinfty(Dn). The quotient H2(Dn)/[h] is an A(Dn) module and the coordinate functions z1, z2, ..., zn act on H2(Dn)/[h] as bounded linear operators. In this paper, we first make a study of the spectral properties of these operators and reveal how these properties are related to the function h. Then we will have a look at the analytic continuation problem. At last, we will show a rigidity phenomenon of quotient Hardy modules.

Zhuan Ye Department of Mathematical Sciences Northern Illinois University, DeKalb, IL 60115 USA (ye@math.niu.edu).
A Unicity Theorem for Meromorphic Mappings, pp. 519-531.
ABSTRACT. We prove a unicity theorem of Nevanlinna for meromorphic mappings of Cn into Pm.

Lang, W. Christopher, Indiana University Southeast, New Albany, Indiana 47150 clang@ius.indiana.edu .
Wavelet analysis on the Cantor dyadic group, pp. 533-544.
Missing pictures can be found  in Vol. 24, No. 4, 1998.
ABSTRACT. Compactly supported orthogonal wavelets are built on the Cantor dyadic group (the dyadic or 2-series local field). Necessary and sufficient conditions are given on a trigonometric polynomial scaling filter for a multiresolution analysis to result. A Lipschitz regularity condition is defined and an unconditional Lp-convergence result is given for regular wavelet expansions (p > 1). Wavelets are given whose scaling filter is a trigonometric polynomial with 2n many terms; regular wavelets with filters with 8 terms are detailed. These wavelets are identified with certain Walsh series on the real line. A Mallat tree algorithm is given for the wavelets.

James C. Alexander, Department of Mathematics, University of Maryland College Park, College Park, MD 20742-4015, USA (jca@math.umd.edu) and Thomas I. Seidman, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250, USA (seidman@math.umbc.edu).
Sliding Modes in Intersecting Switching Surfaces, I: Blending, pp. 545-569.
ABSTRACT. When a flow, discontinuous across a switching surface, points `inward' so one cannot leave, it induces a unique flow within the surface, called the sliding mode. When several such surfaces intersect, one would seek a flow within the intersection, but some difficulties arise. We explore here the extent of the ambiguity involved in this situation and then show that for a certain form of `natural mechanism for implementation' (sigmoid blending) one does indeed inherit, as a residual effect of this implementation, sufficient information to characterize a well-defined sliding mode in the intersection of two switching surfaces.