Rational solitons of wave resonant-interaction models

Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear Schrödinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic construction relies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rational (mixed rational-exponential) solutions and find a broad family of such solutions of the three wave resonant interaction equations

Higher dimensional Automorphic Lie Algebras

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on one hand a powerful tool from the computational point of view, on the other it opens new questions from an algebraic perspective, which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that Automorphic Lie Algebras associated to the $\mathbb{T}\mathbb{O}\mathbb{Y}$ groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only, thus they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.Comment: 43 pages, standard LaTeX2

Automorphic Lie Algebras and Cohomology of Root Systems

A cohomology theory of root systems emerges naturally in the context of Automorphic Lie Algebras, where it helps formulating some structure theory questions. In particular, one can find concrete models for an Automorphic Lie Algebra by integrating cocycles. In this paper we define this cohomology and show its connection with the theory of Automorphic Lie Algebras. Furthermore, we discuss its properties: we define the cup product, we show that it can be restricted to symmetric forms, that it is equivariant with respect to the automorphism group of the root system, and finally we show acyclicity at dimension two of the symmetric part, which is exactly what is needed to find concrete models for Automorphic Lie Algebras. Furthermore, we show how the cohomology of root systems finds application beyond the theory of Automorphic Lie Algebras by applying it to the theory of contractions and filtrations of Lie algebras. In particular, we show that contractions associated to Cartan $\mathbb{Z}$-filtrations of simple Lie algebras are classified by $2$-cocycles, due again to the vanishing of the symmetric part of the second cohomology group.Comment: 26 pages, standard LaTeX2

On the Classification of Automorphic Lie Algebras

It is shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl2-Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.Comment: 29 pages, 1 diagram, 9 tables, standard LaTeX2e, submitted for publicatio

Polyhedral Groups in G2(C)G_2(\mathbb{C})

We classify embeddings of the finite groups $A_4$, $S_4$ and $A_5$ in the Lie group $G_2(\mathbb{C})$ up to conjugation.Comment: 6 pages. To appear in the Glasgow Mathematical Journa

Antigeni salivari quali strumenti epidemiologici per la valutazione dell'esposizione umana ad Aedes albopictus

Hematophagous arthropods during feeding inject into their hosts a cocktail of salivary proteins whose main role is to allow for an effective blood meal by counteracting host hemostasis, inflammation and immunity. However, saliva of blood feeders also evokes in vertebrates an antibody response that can be used to evaluate exposure to disease vectors. Salivary transcriptome studies carried out in different hematophagous species in the last fifteen years clarified the complexity of the salivary repertoires of blood feeding arthropods, pointing out that salivary proteins evolve at a fast evolutionary rate and highlighting the existence of family-, genus- and sometime even species-specific salivary proteins. Focusing on mosquitoes of the genera Anopheles and Aedes, which are important vectors of the human malaria parasite Plasmodium falciparum and of several arboviruses, we summarize here recent efforts to exploit genus-specific salivary proteins as biomarkers of human exposure to these vectors of large relevance for public health

Automorphic Lie Algebras with dihedral symmetry

The concept of Automorphic Lie Algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever-Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is $\mathfrak{sl}_2(\mathbb{C})$ and the poles of the Automorphic Lie Algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In the present research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of Automorphic Lie Algebras with dihedral symmetry, valid for poles at exceptional and generic orbits.Comment: 20 pages, 5 tables, standard LaTeX2

Exact solutions of the 3-wave resonant interaction equation

The Darboux--Dressing Transformations are applied to the Lax pair associated to the system of nonlinear equations describing the resonant interaction of three waves in 1+1 dimensions. We display explicit solutions featuring localized waves whose profile vanishes at the spacial boundary plus and minus infinity, and which are not pure soliton solutions. These solutions depend on an arbitrary function and allow to deal with collisions of waves with various profiles.Comment: 15 pages, 9 figures, standard LaTeX2e, submitted for publication to Physica

Integrability and linear stability of nonlinear waves

It is well known that the linear stability of solutions of (Formula presented.) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general (Formula presented.) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for (Formula presented.) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants

Towards a Solution to Create, Test and Publish Mixed Reality Experiences for Occupational Safety and Health Learning: Training-MR

Artificial intelligence, Internet of Things, Human Augmentation, virtual reality, or mixed reality have been rapidly implemented in Industry 4.0, as they improve the productivity of workers. This productivity improvement can come largely from modernizing tools, improving training, and implementing safer working methods. Human Augmentation is helping to place workers in unique environments through virtual reality or mixed reality, by applying them to training actions in a totally innovative way. Science still has to overcome several technological challenges to achieve widespread application of these tools. One of them is the democratisation of these experiences, for which is essential to make them more accessible, reducing the cost of creation that is the main barrier to entry. The cost of these mixed reality experiences lies in the effort required to design and build these mixed reality training experiences. Nevertheless, the tool presented in this paper is a solution to these current limitations. A solution for designing, building and publishing experiences is presented in this paper. With the solution, content creators will be able to create their own training experiences in a semiassisted way and eventually publish them in the Cloud. Students will be able to access this training offered as a service, using Microsoft HoloLens2. In this paper, the reader will find technical details of the Training-MR, its architecture, mode of operation and communicatio