Convergence of normal form transformations: The role of symmetries

We discuss the convergence problem for coordinate transformations which take a given vector field into Poincar\'e-Dulac normal form. We show that the presence of linear or nonlinear Lie point symmetries can guaranteee convergence of these normalizing transformations, in a number of scenarios. As an application, we consider a class of bifurcation problems.Comment: 20 pages, no figure

Side conditions for ordinary differential equations

We specialize Olver's and Rosenau's side condition heuristics for the determination of particular invariant sets of ordinary differential equations. It turns out that side conditions of so-called LaSalle type are of special interest. Moreover we put side condition properties of symmetric and partially symmetric equations in a wider context. In the final section we present an application to parameter-dependent systems, in particular to quasi-steady state for chemical reactions.Comment: To appear in J. of Lie Theor

Mechanisms producing different precipitation patterns over north‐eastern Italy: insights from HyMeX‐SOP1 and previous events

During the first HyMeX Special Observation Period (SOP1) field campaign, the target site of north‐eastern Italy (NEI) experienced a large amount of precipitation, locally exceeding the climatological values and distributed among several heavy‐rainfall episodes. In particular, two events that occurred during the last period of the campaign drew our attention. These events had common large‐scale patterns and a similar mesoscale setting, characterised by southerly low‐level flow interacting with the Alpine orography, but the precipitation distribution was very different. During Intensive Observing Period IOP18 (31 October–1 November 2012), convective systems were responsible for intense rainfall mainly located over a flat area of the eastern Po Valley, well upstream of the orography. Conversely, during IOP19 (4/5 November 2012), heavy precipitation affected only the Alpine area. In addition to IOP18 and IOP19, the present study analyses other heavy‐precipitation episodes that display similar characteristics and which occurred over NEI during the autumn of recent years. A high‐resolution (2 km grid spacing) non‐hydrostatic NWP model and available observations are used for this purpose. The two different observed precipitation patterns are explained in terms of interaction between the impinging flow and the Alps. Depending on the thermodynamic profile, convection can be triggered when the impinging flow is forced to rise over a pre‐existing cold‐air layer at the base of the orography. In this situation a persistent blocked‐flow condition and upstream convergence are responsible for heavy rain localized over the plain. Conversely, if convection does not develop, flow‐over conditions are established and heavy rain affects the Alps. Numerical parameters proposed in the literature are used to support the analysis. Finally, the role of evaporative cooling beneath the convective systems is evaluated. It turns out that the stationarity of the systems upstream of the Alps is mainly attributable to persistent blocked‐flow conditions, while convective outflow slightly modifies the location of precipitation

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.

Antioxidant and UV-Blocking Functionalized Poly(Butylene Succinate) Films

The introduction of a limited number of functional groups on poly(butylene succinate) (PBS) chains by covalent bonding can impart new properties to the polymer without modifying its thermal and mechanical properties. In pursuit of a viable approach to obtain light- and heat-stabilized PBS samples, the nitroxide radical coupling (NRC) reaction between PBS macroradicals and the 3,5-di-tert-butyl-4-hydroxybenzoyl-2,2,6,6-tetramethylpiperidine-1-oxyl radical (BHB-TEMPO), a functionalizing agent bearing a sterically-hindered antioxidant phenol moiety, is here proposed. The reaction was initiated by peroxide and carried out in solution and in a melt. The functionalized materials were characterized by UV-visible spectroscopy (UV-Vis), proton nuclear magnetic resonance (1H-NMR), and size exclusion chromatography (SEC) analysis to gain structural information and by thermal gravimetric analysis (TGA) and differential scanning calorimetry (DSC) to investigate the thermal properties. In addition, films of the samples were subjected to thermal and photo-oxidative aging to assess their resistance to degradative processes. Finally, the PBS film with the highest degree of functionalization showed the ability to protect β-carotene, a molecule found in food and drugs and that is very sensitive to UV light, from degradation. This result suggests the use of this material (either alone or blended with other biopolyesters) for biodegradable and compostable active packaging

“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

Orbital reducibility and a generalization of lambda symmetries

Abstract We review the notion of reducibility and we introduce and discuss the notion of orbital reducibility for autonomous ordinary differential equations of first order. The relation between (orbital) reducibility and (orbital) symmetry is investigated and employed to construct (orbitally) reducible systems. By standard identifications, the notions extend to nonautonomous ODEs of first and higher order. Moreover we thus obtain a generalization of the lambda symmetries of Muriel and Romero. Several examples are given

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

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