Volume 28, Number 1, 2002
COMPACTNESS IN THE FIRST BAIRE CLASS AND BAIRE-1 OPERATORS
S. Mercourakis email@example.com, E. Stamati
2000 Mathematics Subject Classification: Primary 46A50, 46E40, 47B99; Secondary 54C35. Key words: Baire-1 function, Baire-1 operator, Rosenthal compact, Rosenthal-Banach compact, polish space, angelic space, Bounded approximation property.
In the second part of the paper we examine under which conditions a bounded linear operator T:X*® Y so that T|BX*: (BX*,w*)® Y is a Baire-1 function, is a pointwise limit of a sequence (Tn) of operators with T|BX* :(BX*,w*) ® (Y, ||.||) continuous for all n Î N. Our results in this case are connected with classical results of Choquet, Odell and Rosenthal.
POROSITY AND VARIATIONAL PRINCIPLES
Elsa M. Marchini firstname.lastname@example.org
2000 Mathematics Subject Classification: 49K40, 49J99, 54E52. Key words: Variational principles, well-posed optimization problems, porous sets, porosity.
ON PARABOLIC SUBGROUPS AND HECKE ALGEBRAS OF SOME FRACTAL GROUPS
Laurent Bartholdi email@example.com, Rostislav I. Grigorchuk firstname.lastname@example.org
2000 Mathematics Subject Classification: 20F50, 20C12. Key words: Branch Group; Fractal Group; Parabolic Subgroup; Quasi-regular Representation; Hecke Algebra; Gelfand Pair; Growth; L-Presentation; Tree-like Decomposition.
We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets of the parabolic subgroup. Since our results derive from considerations on finite-index subgroups, they also hold for the profinite completions ^G of the groups G.
The representations involved have interesting spectral properties investigated in . This paper serves as a group-theoretic counterpart to the studies in the mentioned paper.
We study more carefully a few examples of fractal groups, and in doing so exhibit the first example of a torsion-free branch just-infinite group.
We also produce a new example of branch just-infinite group of intermediate growth, and provide for it an L-type presentation by generators and relators.
MULTIBODY SYSTEM MECHANICS: MODELLING, STABILITY, CONTROL, AND ROBUSTNESS,
by V. A. Konoplev and A. Cheremensky email@example.com,
Mathematics and its Applications Vol. 1, Union of Bulgarian Mathematicians, Sofia, 2001, XXII+288 pp., $ 65.00, ISBN 954-8880-09-01
2000 Mathematics Subject Classification: 93-02, 93B51, 93C83. Key words: Stability, Control, Robustness, Multibody systems, Computer-aided mathematical formalism, System design, Modelling.