In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 × 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU (2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP 2 = F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.

Purpose Odontogenic sinusitis and sinonasal complications of dental disease or treatment (SCDDT) represent a heterogeneous group of conditions that often require multidisciplinary care. The present study aims to prospectively validate a classification and treatment protocol for SCDDT patients. Methods One hundred twenty-eight consecutive patients (73 females and 45 males, mean age 52.4 years) affected by SCDDT not responding to dental and medical therapy were classified and surgically treated according to the proposed protocol. The protocol classified patients into three aetiology-based groups (preimplantologic, implantologic, and related to traditional dental diseases and procedures, respectively). The groups were further divided into classes according to the presence of oro-antral communications and/or dislocated dental hardware. Each condition was treated according to the class-related, protocol-defined treatment, by either a transnasal or combined transnasal/transoral approach. All patients were successfully classified according to our protocol. None of the proposed classes were redundant, and no condition fell outside the definitions. Results The surgical treatment protocol proved to be adequate and effective, in that 125 of the 128 patients completely recovered after surgical treatment. Conclusions The term SCDDT and the consequent classification proposed by the authors appear, therefore, to be nosologically correct. Furthermore, the protocol-related proposed treatment appears to be clinically sound, with a success rate nearing 98%.

We provide a simple coordinatization for the group G2, which is analogous to the Euler coordinatization for SU (2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H. * cacciatori@mi.infn.it † BLCerchiai@lbl.gov ‡ alberto.dellavedova@unimib.it § giovanni.ortenzi@unimib.it ¶ Graduate visitor (visitatore laureato), ascotti@mindspring.com

The classical equations of motion of a one-dimensional, finite, anharmonic lattice, with nearest-neighbor interaction of the Lennard-Jones type, are investigated numerically.The results indicate that when the vibrational energy per particle is equal to or greater than 2-3% of the depth of the potential well, one has, in time average, equipartition of the energy among the normal modes, thus giving a hint toward ergodicity of the system at sufficiently high energy.For lower energies one finds recurrent motions if initially only one normal mode is excited in analogy with a famous result due to Fermi, Pasta, and Ulam. In this case the numerical results are consistent both with the existence of a long relaxation time and with a lack of ergodicity for low energies.