An approach to Mel'nikov theory in celestial mechanics

Using a completely analytic procedure - based on a suitable extension of a classical method - we discuss an approach to the Poincar\'e-Mel'nikov theory, which can be conveniently applied also to the case of non-hyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described by Kepler-like potentials in one or two degrees of freedom, in the presence of general time-dependent perturbations. We show that the appearance of chaos (possibly including Arnol'd diffusion) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee or blowing-up transformations. Natural examples are provided by the classical Gyld\'en problem, originally proposed in celestial mechanics, but also of interest in different fields, and by the general 3-body problem in classical mechanics.Comment: LaTeX, no figure

On the relation between standard and μ\mu-symmetries for PDEs

We give a geometrical interpretation of the notion of $\mu$-prolongations of vector fields and of the related concept of $\mu$-symmetry for partial differential equations (extending to PDEs the notion of $\lambda$-symmetry for ODEs). We give in particular a result concerning the relationship between $\mu$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $\mu$-symmetries and nonlocal standard symmetries.Comment: 25 pages, no figures, latex. to be published in J. Phys.

Poincare' normal forms and simple compact Lie groups

We classify the possible behaviour of Poincar\'e-Dulac normal forms for dynamical systems in $R^n$ with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and thus the corresponding simple compact Lie groups. The ``renormalized forms'' (in the sense of previous work by the author) of these systems is also discussed; in this way we are able to simplify the classification and moreover to analyze systems with zero linear part. We also briefly discuss the convergence of the normalizing transformations.Comment: 17 pages; minor corrections in revised versio

Preliminary study on MC1R polymorphism in some cattle breeds raised in Italy

Ricerche preliminari sul polimorfismo del gene MC1R in alcune razze bovine allevate in Italia – Il gene MC1R è stato analizzato in 193 soggetti appartenenti a 8 razze bovine, tramite PCR, per la presenza di due mutazioni ad effetto fenotipico noto sulla pigmentazione del mantello: la delezione G310 e la sostituzione T296C, associate rispettivamente al fenotipo feomelanico (e) ed eumelanico nero (Ed). Sessanta soggetti di razza Limousine e Pezzata Rossa Italiana presentano genotipo e/e; 27 soggetti di razza Frisona Italiana mostrano genotipo Ed/Ed mentre 2 genotipo Ed/e. Gli 84 soggetti appartenenti alle razze Cabannina, Chianina, Marchigiana e Piemontese non presentano tali mutazioni, analogamente a 18 soggetti di razza Romagnola, nella quale però si sono anche osservati 2 soggetti portatori dell’allele e allo stato eterozigote

“Intestinal-Type” Vulvar Adenocarcinoma: A Review of the MITO Rare Tumors Group

Intestinal-type adenocarcinoma (VAIt) represents a sporadic variant of vulvar carcinoma. It appears frequently localized to epithelial glands in the vulvar region, and it probably derives from cloacal remnants persisting in the adult. We performed a systematic review of the limited cases reported in the literature, with the intent to assess the specific peculiarities of this rare neoplasia and to state consistent management recommendations. The principal histological VAIt characteristic is that it resembles mucinous colonic carcinomas. Therefore, immunohistochemical workup, with different tumor markers including CK20, CDX2, and CK7 staining, is needed. To confirm vulvar origin, a thorough diagnostic, and radiological examination is required to rule out other primary malignancies. The gold standard of treatment for VAIt is surgery, with local excision with tumor-free margins. Lymph node staging is an option advised if the tumor size is >2 cm or if lymph node metastases are suspected on imaging. On the other hand, the role of neoadjuvant therapy is still in doubt, but a good response to adjuvant chemotherapy treatments has been described in both advanced and recurrent diseases. Sometimes, VAIt behavior can be unpredictable, with relapses even after many years, so more experiences and longer follow-up periods are needed to elucidate the best therapeutic management and its long-term prognosis

Local and nonlocal solvable structures in ODEs reduction

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure

On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions

Weak Transversality and Partially Invariant Solutions

New exact solutions are obtained for several nonlinear physical equations, namely the Navier-Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schroedinger equation. The solution methods make use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.Comment: 23 pages, preprint CRM-284