Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems

We consider Hamiltonian systems near equilibrium that can be (formally) reduced to one degree of freedom. Spatio-temporal symmetries play a key role. The planar reduction is studied by equivariant singularity theory with distinguished parameters. The method is illustrated on the conservative spring-pendulum system near resonance, where it leads to integrable approximations of the iso-energetic Poincaré map. The novelty of our approach is that we obtain information on the whole dynamics, regarding the (quasi-) periodic solutions, the global configuration of their invariant manifolds, and bifurcations of these.

Transformations of Feynman path integrals and generalized densities of Feynman pseudomeasures

Applications of transformations of Feynman path integrals and Feynman pseudomeasures to explain arising quantum anomalies are considered. A contradiction in the literature is also explained

Point vortices on the sphere: a case with opposite vorticities

We study systems formed of 2N point vortices on a sphere with N vortices of strength +1 and N vortices of strength -1. In this case, the Hamiltonian is conserved by the symmetry which exchanges the positive vortices with the negative vortices. We prove the existence of some fixed and relative equilibria, and then study their stability with the ``Energy Momentum Method''. Most of the results obtained are nonlinear stability results. To end, some bifurcations are described.Comment: 35 pages, 9 figure

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page

Stability of relative equilibria with singular momentum values in simple mechanical systems

A method for testing $G_\mu$-stability of relative equilibria in Hamiltonian systems of the form "kinetic + potential energy" is presented. This method extends the Reduced Energy-Momentum Method of Simo et al. to the case of non-free group actions and singular momentum values. A normal form for the symplectic matrix at a relative equilibrium is also obtained.Comment: Partially rewritten. Some mistakes fixed. Exposition improve

Centralized cytogenetic analysis of pediatric acute leukemia: results of an italian collaborative experience

Background and Objective. Cytogenetic analysis of acute leukemia yields important information which has been demonstrated to be correlated to patient survival. A reference laboratory was created in order to perform karyotype analysis on all cases of acute leukemia enrolled in the AIEOP (Associazione Italiana Emato-Oncologia Pediatrica) protocols. Methods. From January 1990 to December 1995, 1115 samples of children with ALL or AML were sent in for cytogenetic analysis. The results of cell cultures were screened in the Reference Laboratory and then the fixed metaphases were sent to one of the six cytogenetic laboratories for analysis. Results. The leukemic karyotypes of 556 patients were successfully analyzed. An abnormal clone was detected in 49% of cases of ALL and in 66% of AML. In ALL the most frequent abnormality was 9p rearrangement. Other recurrent abnormalities were t(9;22), t(4;11) and t(1;19). In AML t(8;21), t(15;17) and 11q23 rearrangement were the most frequent structural abnormalities. These findings are similar to the results obtained in other multicenter studies using a similar approach. Interpretation and Conclusions. Our data confirm the feasibility of performing cytogenetic analysis in a centralized laboratory on mailed samples of bone marrow and/or peripheral blood; this is very important considering that cytogenetic analysis of neoplastic tissue requires a special laboratory and expert staff

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.

Bethe--Salpeter equation in QCD

We extend to regular QCD the derivation of a confining $q \bar{q}$ Bethe--Salpeter equation previously given for the simplest model of scalar QCD in which quarks are treated as spinless particles. We start from the same assumptions on the Wilson loop integral already adopted in the derivation of a semirelativistic heavy quark potential. We show that, by standard approximations, an effective meson squared mass operator can be obtained from our BS kernel and that, from this, by ${1\over m^2}$ expansion the corresponding Wilson loop potential can be reobtained, spin--dependent and velocity--dependent terms included. We also show that, on the contrary, neglecting spin--dependent terms, relativistic flux tube model is reproduced.Comment: 23 pages, revte