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Product Code: Vol. 37, No. 1, 2011

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**McNair, Dawn B.,** Johnson C. Smith University, Charlotte, NC 28216 (dmcnair@jcsu.edu).

Duals of ideals in rings with zero divisors, pp. 1-26.

ABSTRACT For any nonzero ideal I of ring R, we define the inverse of I as the set of elements from Q(R) (complete ring of quotients of R) that conducts I into R and call it the dual of I . Much work has been done with regard to determining when the dual of I is a ring in the case R is an integral domain. This paper will extend those results to dense ideals in rings with zero divisors. Attention will also be given to duals of ideals in Prüfer and strong Prüfer rings.

**Trotta, Belinda**, La Trobe University, Victoria 3086, Australia (belindatrotta@gmail.com).

Residual properties of reflexive, anti-symmetric digraphs, pp. 27-46.

ABSTRACT. For a quasivariety C of reflexive anti-symmetric digraphs, we consider the class RCT(Cfin) of topological digraphs that are topologically residually in the class of finite (discretely topologised) members of C. In particular, we show that RCT(Cfin) can be axiomatised (among compact, totally disconnected digraphs) by first-order sentences if and only if C is strictly contained within the quasivariety of partial orders. This work extends Stralka's result that there is a compact, totally disconnected partially ordered space that is not a Priestley space.

**Mabrouk Ben Nasr, ** and **Noômen Jarboui**, King Faisal University, Faculty Of Sciences, Department Of Mathematics, P. O. Box: 380, 31982 Saudi Arabia.(mabrouk_bennasr@yahoo.fr) , (noomenjarboui@yahoo.fr).

On maximal non-valuation subrings, pp. 47-59.

ABSTRACT. In this paper we study pairs of rings where all intermediate domains are valuation. Furthermore maximal non-valuation subrings are studied and examples illustrating the theory are given.

**Martín Méndez,** Alberto, Universidade de Vigo, 36310, Vigo (Pontevedra), Spain (amartin@dma.uvigo.es), and **Torres Lopera,** **Juan Francisco,** Universidade de Santiago de Compostela, 15706, Santiago de Compostela (La Coruña), Spain (jftorre@usc.es).

Tensorial structures associated with semisimple graded Lie algebras, pp. 61-77.

ABSTRACT. Properties concerning several tensors associated with geometric structures of graded type are studied. Particularly, we study Tanaka tensor and Weyl curvature tensor and some relations between them. We prove that the difference tensor of two linear connections on a manifold endowed with a geometric structure of graded type is null if and only if the connections have the same torsion. An explicit calculus of Tanaka tensor for classical simple real graded Lie algebras is given.

**Dethloff, Gerd, **University of Brest, 29275 Brest, France (dethloff@univ-brest.fr), and ** Tan, Tran Van, **ENS Hanoi, Hanoi, Vietnam .

A second Main Theorem for moving hypersurface targets, pp. 79-111.

ABSTRACT. In this paper, we prove a Second Main Theorem for algebraically nondegenerate meromorphic maps of C^m into CP^n with at least n+2 slowly moving hypersurface targets in (weakly) general position. We also introduce a truncation, with an explicit estimate of the truncation level, into this Second Main Theorem. This generalizes recent works of Min Ru and An-Phuong on Second Main Theorems for fixed hypersurface targets.

**Cohen, Nir, **University of Campinas, Campinas, Brazil, **Grama, Lino, ** University of Campinas, Campinas, Brazil, and ** Negreiros, Caio J.C., **University of Campinas, Campinas, Brazil (caione@ime.unicamp.br).

Equigeodesics on flag manifolds, pp. 113-125.

ABSTRACT. This paper provides a characterization of homogeneous curves on a geometric flag manifold which are geodesics with respect to each invariant metric. We call such curves homogeneous equigeodesics. We also characterize homogeneous equigeodesics whose associated Killing field is closed, hence, the corresponding geodesics are closed.

**Biliotti, Leonardo,** Dipartimento di Matematica, Università di Parma, Via G. Usberti, 53/A 43100, Parma, Italy (leonardo.biliotti@unipr.it), and **Javaloyes, Miguel Angel,** Departamento de Geometrí a y Topologíca, Universidad de Granada, Campus Fuentenueva S/N, 18071, Granada, Spain (majava@ugr.es)

t-periodic light rays in conformally stationary spacetimes via Finsler geometry,pp. 127-146.

ABSTRACT. In this paper we prove several multiplicity results of t-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of t-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is η-pinched.

**SuceavÄ, Bogdan D.**, California State University at Fullerton, Fullerton, CA 92834-6850 (bsuceava@fullerton.edu).

Distances generated by Barbilian's metrization procedure by oscillation of sublogarithmic functions, pp. 147-159.

ABSTRACT. Introduced originally in 1934, Barbilian’s metrization procedure induced a distance on a planar domain by a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to domains of a more general form, withstanding not necessarily on planar sets, but in a more abstract setting. In the present work, we show that there exists more general classes of distances than the ones produced by logarithmic oscillation. As a consequence, in Theorem 2 we state the most general form of Barbilian’s metrization procedure.

**Anselm Knebusch, **Department of Mathematics, Bunsenstrasse 3-5, D-37073 Göttingen, Germany (knebusch@math.uni-goettingen.de).

Approximation of center-valued Betti-numbers, pp. 161-179.

ABSTRACT. In this paper we generalize the ordinary approximation theorem to calculate L^{2}-Betti-numbers to an approximation theorem for universal Betti-numbers.

**Cabello Sanchez, Felix,** UEx, Badajoz 06071, Spain (fcabello@unex.es), and **Cabello Sanchez,** **Javier,** UEx, Badajoz 06071, Spain (coco@unex.es)

Nonlinear isomorphisms of lattices of Lipschitz functions, pp. 181-202.

ABSTRACTThe paper contains a number of Banach-Stone type theorems for lattices of uniformly continuous and Lipschitz functions without any linearity assumption. Sample result: two complete metric spaces of finite diameter are Lipschitz homeomorphic if (and only if, of course) the corresponding lattices of Lipschitz functions are isomorphic. Here, a lattice isomorphism is just a bijection preserving the order in both directions, in particular linearity is not assumed.

**Shalit, Orr M., **Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L-3G1, CANADA (oshalit@math.uwaterloo.ca).

E-dilation of strongly commuting CP-semigroups (the nonunital case), pp. 203-232.

ABSTRACT. In a previous paper, we showed that every strongly commuting pair of CP0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to a two-parameter contraction semigroup.

**Victor Kaftal** and **Gary Weiss**, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA (kaftal@math.uc.edu) and (gary.weiss@math.uc.edu).

B(H) lattices, density and arithmetic mean ideals, pp. 233-283.

ABSTRACT. Lattice properties of operator ideals in B(H) with applications to the arithmetic mean ideals introduced by Dykema, Figiel, Weiss and Wodzicki (Adv Math 2004) are studied here as part of a five paper project announced in PNAS 2002. We focus on the general lattice of B(H)-ideals and on particular sublattices such as the principal and countably generated ideals and their density properties (between any ideal and an ideal in a sublattice lies another ideal in that sublattice). As applications, we obtain cancellation properties for first order arithmetic mean ideals and arithmetic mean ideals at infinity and solve related ideal optimization problems.

**Klemes, Ivo**, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, H3A 2K6, Canada (klemes@math.mcgill.ca).

Symmetric polynomials and lp inequalities for certain intervals of p, pp. 285-295.

ABSTRACT. We prove some sufficient conditions implying lp inequalities of the form ||x||p ≤ ||y||p for vectors x, y in Rn and for p in certain positive real intervals. Our sufficient conditions are strictly weaker than the usual majorization relation. The conditions are expressed in terms of certain homogeneous symmetric polynomials in the entries of the vectors. These polynomials include the elementary symmetric polynomials as a special case. We also give a characterization of the majorization relation by means of symmetric polynomials.

**Li, Haigang,** School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R. China, and **Bao, Jiguang,** School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R. China (jgbao@bnu.edu.cn).

Existence of rotating stars with prescribed angular velocity law, pp. 297-309.

ABSTRACT. The existence of solutions of the equations for a self-gravitating fluid with prescribed angular velocity law is proved. The conditions on the angular velocity are nearly optimal. The system is formulated as a variational problem and concentration-compactness methods are used to prove the existence of minimizers of the energy functional.

**Zhao Dongsheng,** Mathematics and Mathematics Education, National Institute of Education Singapore, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616(dongsheng.zhao@nie.edu.sg).

A partial order on the set of continuous endomappings, pp. 311-326.

ABSTRACT. Motivated by the reflexive operator algebra problem, we introduce a new partial order on the set End_{C}(X) of all continuous endomappings on a topological space X. We study the relationship between the topological structure and the order structure, and we establish conditions for certain families of continuous endomappings to be reflexive. We introduce the V-regular spaces and show that if X and Y are V-regular spaces, then the two semigroups End_{C}(X) and End_{C}(Y) are isomorphic if and only if X and Y$ are homeomorphic.

**SpÄvák**, **Jan**, Department of mathematics, J.E. Purkinje University, Ceske mladeze 8, 400 96 Usti nad Labem, Czech Republik (jan.spevak@ujep.cz).

Finite-valued mappings preserving dimension, pp. 327-348.

ABSTRACT. We say that a set-valued mapping F: X⇒Y is *C*-lsc provided that there exists a countable cover *C* of X consisting of functionally closed sets such that for every C∈*C* and each functionally open subset U of Y one can find a functionally open set V⊂X such that {x∈C: F(x)∩ U≠Ø}=C∩V. For Tychonoff spaces X and Y we say that X dominates Y provided that there exist a finite-valued *C*-lsc mapping F: X⇒Y and a finite-valued *D*-lsc mapping G:Y⇒X (for suitable *C* and *D*) such that y∈ ∪{F(x):x∈G(y)} for every y∈Y. We prove that if X dominates Y, then dim X≥dim Y. (Here dim X denotes the Äech-Lebesgue (covering) dimension of X.) As a corollary, we obtain that dim X=dim Y whenever a perfectly normal space Y is an image of a Tychonoff space X under a finite-to-one open mapping. We also give an example of an open mapping f:X→Y such that |f^{-1}(y)|≤2 for all y∈Y, both X and Y are hereditarily normal (and Y is even Lindelöf) but dim X≠dim Y.

**Editors Addendum **on

Singularities of generic lightcone Gauss maps and lightcone pedal surfaces of spacelike curves in Minkowski 4-space by *L.L. Kong, R.M. Gao, D.H. Pei, and J.H. Zhang.*

After the paper had already appeared in print, HJM Vol. 36(3) pp. 697-710, the referee discovered that the paper had an irreparable error. Proposition 2.1 in the paper is not correct. From that place on, all the arguments fail and thus the proof of Theorem B is not correct. This has been confirmed by the authors.

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