Six types of E−E-functions of the Lie groups O(5) and G(2)

New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently found in the literature. All the families are fully characterized, their most important properties are described, namely their continuous and discrete orthogonalities and decompositions of their products.Comment: 25 pages, 13 figure

Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation

The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that whenever they have the Painlev\'e property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio

Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.Comment: 16 pages, LaTeX, no figure

Recursion relations and branching rules for simple Lie algebras

The branching rules between simple Lie algebras and its regular (maximal) simple subalgebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights $\Gamma$ specific for each injection. The structure of $\Gamma$ is easily deduced from the correspondence between the root systems of algebra and subalgebra. The recursion relations thus obtained give rise to simple and effective algorithm for branching rules. The details are exposed by performing the explicit decomposition procedure for $A_{3} \oplus u(1) \to B_{4}$ injection.Comment: 15p.,LaTe

Maximal Abelian Subgroups of the Isometry and Conformal Groups of Euclidean and Minkowski Spaces

The maximal Abelian subalgebras of the Euclidean e(p,0) and pseudoeuclidean e(p,1)Lie algebras are classified into conjugacy classes under the action of the corresponding Lie groups E(p,0) and E(p,1), and also under the conformal groups O(p+1,1) and O(p+1,2), respectively. The results are presented in terms of decomposition theorems. For e(p,0) orthogonally indecomposable MASAs exist only for p=1 and p=2. For e(p,1), on the other hand, orthogonally indecomposable MASAs exist for all values of p. The results are used to construct new coordinate systems in which wave equations and Hamilton-Jacobi equations allow the separation of variables.Comment: 31 pages, Latex (+ latexsym

Extended calibration range for prompt photon emission in ion beam irradiation

Monitoring the dose delivered during proton and carbon ion therapy is still a matter of research. Among the possible solutions, several exploit the measurement of the single photon emission from nuclear decays induced by the irradiation. To fully characterize such emission the detectors need development, since the energy spectrum spans the range above the MeV that is not traditionally used in medical applications. On the other hand, a deeper understanding of the reactions involving gamma production is needed in order to improve the physic models of Monte Carlo codes, relevant for an accurate prediction of the prompt-gamma energy spectrum.This paper describes a calibration technique tailored for the range of energy of interest and reanalyzes the data of the interaction of a 80MeV/u fully stripped carbon ion beam with a Poly-methyl methacrylate target. By adopting the FLUKA simulation with the appropriate calibration and resolution a significant improvement in the agreement between data and simulation is reported.Comment: 4 pages, 7 figures, Submitted to JINS

Solvable Lie algebras with triangular nilradicals

All finite-dimensional indecomposable solvable Lie algebras $L(n,f)$, having the triangular algebra T(n) as their nilradical, are constructed. The number of nonnilpotent elements $f$ in $L(n,f)$ satisfies $1\leq f\leq n-1$ and the dimension of the Lie algebra is $\dim L(n,f)=f+{1/2}n(n-1)$

Quantum transfer matrices for discrete and continuous quasi-exactly solvable problems

We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them into the framework of the quantum inverse scattering method. The quasi-exactly solvable hamiltonians in one dimension are identified with traces of quantum monodromy matrices for specific integrable systems with non-periodic boundary conditions. Applications to the Azbel-Hofstadter problem are outlined.Comment: 15 pages, standard LaTe

Measurement of charged particle yields from therapeutic beams in view of the design of an innovative hadrontherapy dose monitor

Particle Therapy (PT) is an emerging technique, which makes use of charged particles to efficiently cure different kinds of solid tumors. The high precision in the hadrons dose deposition requires an accurate monitoring to prevent the risk of under-dosage of the cancer region or of over-dosage of healthy tissues. Monitoring techniques are currently being developed and are based on the detection of particles produced by the beam interaction into the target, in particular: charged particles, result of target and/or projectile fragmentation, prompt photons coming from nucleus de-excitation and back-to-back γ s, produced in the positron annihilation from β + emitters created in the beam interaction with the target. It has been showed that the hadron beam dose release peak can be spatially correlated with the emission pattern of these secondary particles. Here we report about secondary particles production (charged fragments and prompt γ s) performed at different beam and energies that have a particular relevance for PT applications: 12C beam of 80 MeV/u at LNS, 12C beam 220 MeV/u at GSI, and 12C, 4He, 16O beams with energy in the 50–300 MeV/u range at HIT. Finally, a project for a multimodal dose-monitor device exploiting the prompt photons and charged particles emission will be presented

Realizations of Real Low-Dimensional Lie Algebras

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in Appendix are correcte