Serdica Mathematical Journal

Volume 28, Number 1, 2002

C O N T E N T S

Articles

• Mercourakis, S., E. Stamati. Compactness in the first Baire class and Baire-1 operators (pp. 1-36)
• Marchini, E. M. Porosity and variational principles (pp. 37-46)
• Bartholdi, L., R. I. Grigorchuk. On parabolic subgroups and Hecke algebras of some fractal groups (pp. 47-90)

  Book Reviews

• Grozdev, S. (Reviewer) Multibody System Mechanics: Modelling, Stability, Control, and Robustness by V. A. Konoplev and A. Cheremensky (pp. 91-93)

COMPACTNESS IN THE FIRST BAIRE CLASS AND BAIRE-1 OPERATORS
S. Mercourakis smercour@math.uoa.gr, 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.

For a polish space M and a Banach space E let B₁(M,E) be the space of first Baire class functions from M to E, endowed with the pointwise weak topology. We study the compact subsets of B₁(M,E) and show that the fundamental results proved by Rosenthal, Bourgain, Fremlin, Talagrand and Godefroy, in case E = R, also hold true in the general case. For instance: a subset of B₁(M,E) is compact iff it is sequentially (resp. countably) compact, the convex hull of a compact bounded subset of B₁(M,E) is relatively compact, etc. We also show that our class includes Gulko compact.

In the second part of the paper we examine under which conditions a bounded linear operator T:X^*® Y so that T|_BX^*: (B_X^*,w^*)® Y is a Baire-1 function, is a pointwise limit of a sequence (T_n) of operators with T|_BX^* :(B_X^*,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 elsa@matapp.unimib.it

2000 Mathematics Subject Classification: 49K40, 49J99, 54E52. Key words: Variational principles, well-posed optimization problems, porous sets, porosity.

We prove that in some classes of optimization problems, like lower semicontinuous functions which are bounded from below, lower semicontinuous or continuous functions which are bounded below by a coercive function and quasi-convex continuous functions with the topology of the uniform convergence, the complement of the set of well-posed problems is s-porous. These results are obtained as realization of a theorem extending a variational principle of Ioffe-Zaslavski.

ON PARABOLIC SUBGROUPS AND HECKE ALGEBRAS OF SOME FRACTAL GROUPS
Laurent Bartholdi laurent@math.berkeley.edu, Rostislav I. Grigorchuk grigorch@mi.ras.ru

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 study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type.

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 [6]. 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.

Grozdev, S. savagroz@math.bas.bg (Reviewer)
MULTIBODY SYSTEM MECHANICS: MODELLING, STABILITY, CONTROL, AND ROBUSTNESS,
by V. A. Konoplev and A. Cheremensky ana@bgcict.acad.bg,
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.