On a product formula for the Conley-Zelmder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley-Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

GALAXY EVOLUTION IN A COMPLEX ENVIRONMENT: A MULTI-WAVELENGTH STUDY OF HCG 7

The environment where galaxies are found heavily influences their evolution. Close groupings, like the ones in the cores of galaxy clusters or compact groups, evolve in ways far more dramatic than their isolated counterparts. We have conducted a multi-wavelength study of Hickson Compact Group 7 (HCG 7), consisting of four giant galaxies: three spirals and one lenticular. We use Hubble Space Telescope (HST) imaging to identify and characterize the young and old star cluster populations. We find young massive clusters (YMCs) mostly in the three spirals, while the lenticular features a large, unimodal population of globular clusters (GCs) but no detectable clusters with ages less than a few Gyr. The spatial and approximate age distributions of the similar to 300 YMCs and similar to 150 GCs thus hint at a regular star formation history in the group over a Hubble time. While at first glance the HST data show the galaxies as undisturbed, our deep ground-based, wide-field imaging that extends the HST coverage reveals faint signatures of stellar material in the intragroup medium (IGM). We do not, however, detect the IGM in H I or Chandra X-ray observations, signatures that would be expected to arise from major mergers. Despite this fact, we find that the H I gas content of the individual galaxies and the group as a whole are a third of the expected abundance. The appearance of quiescence is challenged by spectroscopy that reveals an intense ionization continuum in one galaxy nucleus, and post-burst characteristics in another. Our spectroscopic survey of dwarf galaxy members yields a single dwarf elliptical galaxy in an apparent stellar tidal feature. Based on all this information, we suggest an evolutionary scenario for HCG 7, whereby the galaxies convert most of their available gas into stars without the influence of major mergers and ultimately result in a dry merger. As the conditions governing compact groups are reminiscent of galaxies at intermediate redshift, we propose that HCGs are appropriate for studying galaxy evolution at z similar to 1-2.

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation -Delta u = lambda u +/- (x, u) + h(x) in a bounded domain, where f has a sublinear growth and h is an element of L-2. We find suitable conditions on f and It in order to have at least two solutions for X near to an eigenvalue of -Delta. A typical example to which our results apply is when f (x, u) behaves at infinity like a(x)vertical bar u vertical bar(q-2)u, with M > a(x) > delta > 0, and I < q < 2. (C) 2007 Elsevier Inc. All rights reserved.

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system U(A+B)(t, s)0 <= s <= t <= T generated by the sum -(A(t) + B(t)) of two linear, time-dependent, and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express U(A+B)(t, s)0 <= s <= t <= T as the strong limit in C(8) of a product of the holomorphic contraction semigroups generated by -A (t) and - B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t) + B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND(t is an element of)[0,T] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct U(A)(t, s)0 <=(s)<=(t)<=(T) very simply in terms of the semigroup generated by -A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of timedependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrodinger type in quantum mechanics.

Compressibility of electron liquid in the quantized Hall phase of GaAs/AlGaAs multi-layers

Magneto-capacitance was studied in narrow miniband GaAs/AlGaAs superlattices where quasi-two dimensional electrons revealed the integer quantum Hall effect. The interwell tunneling was shown to reduce the effect of the quantization of the density of states on the capacitance of the superlattices. In such case the minimum of the capacitance observed at the filling factor nu = 2 was attributed to the decrease of the electron compressibility due to the formation of the incompressible quantized Hall phase. In accord with the theory this phase was found strongly inhomogeneous. The incompressible fraction of the quantized Hall phase was demonstrated to rapidly disappear with the increasing temperature. (C) 2008 Elsevier B.V. All rights reserved.