Volume 107, Issue 3
August 1997, pages 223-327
pp 223-235 August 1997
Constructing irreducible representations of discrete groups
Marc Burger Pierre De La Harpe
The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups Γ_{0} with Γ its own commensurator in Γ. These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups.
pp 237-242 August 1997
A new proof of Suzuki's formula
We give a different proof of a formula of Suzuki and its strengthening by Zaidenberg for the topological Euler characteristic of an affine surface fibered over a curve. We deduce this formula using the ideas for the proof of an analogous formula for a proper morphism.
pp 243-250 August 1997
Substantial Riemannian submersions ofS^{15} with 7-dimensional fibres
In this paper we show that a substantial Riemannian submersion ofS^{15} with 7-dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of the diameter rigidity theorem for the Cayley plane which is considerably stronger than the very recent radius rigidity theorem of Wilhelm.
pp 251-261 August 1997
Harmonic manifolds with some specific volume densities
We show that a noncompact, complete, simply connected harmonic manifold (M^{d}, g) with volume densityθ_{m}(r)=sinh^{d-1}r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M^{2d}, g) with volume densityθ_{m}(r)=sinh^{2d-1}r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.
pp 263-270 August 1997
Some inequalities for the polar derivative of a polynomial
LetP(z) be a polynomial of degreen which does not vanish in |z|<1. In this paper, we estimate the maximum and minimum moduli of thekth polar derivative ofP(z) on |z|=1 and thereby obtain compact generalizations of some known results, which among other results, yields interesting refinements of Erdos-Lax theorem and a theorem of Ankeny and Rivlin.
pp 271-276 August 1997
A theorem concerning a product of two general classes of polynomials and the multivariableH-function
A theorem concerning a product of two general classes of polynomials and the multivariableH-function is established. Certain integrals and expansion formulae have also been derived by the application of this theorem. This general theorem yields a number of new, interesting and useful theorems, integrals and expansion formulae as its particular cases.
pp 277-282 August 1997
Weakly analytic sets for function spaces
We define and study weakly analytic sets for function spaces. This concept generalizes the concept of weak analyticity for function algebras. We also define weak analyticity for the space of affine functions using the ideas of convexity. We show that it coincides with the same concept regarding it as a function space. We give several examples to illustrate the concepts.
pp 283-317 August 1997
Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR^{2}
Let Ω be a bounded smooth domain inR^{2}. Letf:R→R be a smooth non-linearity behaving like exp{s^{2}} ass→∞. LetF denote the primitive off. Consider the functionalJ:H_{0}^{1}(Ω)→R given by$$J(u) = \frac{1}{2}\int_\Omega {\left| {\nabla u} \right|^2 dx - } \int_\Omega {F(u)dx.} $$ It can be shown thatJ is the energy functional associated to the following nonlinear problem: −Δu=f(u) in Ω,u=0 on ρΩ. In this paper we consider the global compactness properties ofJ. We prove thatJ fails to satisfy the Palais-Smale condition at the energy levels {k/2},k any positive integer. More interestingly, we show thatJ fails to satisfy the Palais-Smale condition at these energy levels along two Palais-Smale sequences. These two sequences exhibit different blow-up behaviours. This is in sharp contrast to the situation in higher dimensions where there is essentially one Palais-Smale sequence for the corresponding energy functional.
pp 319-327 August 1997
From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus
In this article we give a new proof of Ito's formula inR^{n} starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.
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