Variational principles for involutive systems of vector fields

In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar way by means of an higher order variational principle, and how this extends to involutive systems of vector fields.Comment: 31 pages. To appear in International Journal of Geometric Methods in Modern Physics (IJGMMP

Weierstrass's criterion and compact solitary waves

Weierstrass's theory is a standard qualitative tool for single degree of freedom equations, used in classical mechanics and in many textbooks. In this Brief Report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist. In the framework of continuum mechanics, these differential equations correspond to bulk shear waves for a special class of constitutive laws.Comment: 4 page

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

fNIRS neuroimaging in olfactory research: A systematic literature review

There are a number of key features which make olfaction difficult to study; subjective processes of odor detection, discrimination and identification, and individualistic odor hedonic perception and associated odor memories. In this systematic review we explore the role functional near-infrared spectroscopy (fNIRS) has played in understanding olfactory perception in humans. fNIRS is an optical neuroimaging technique able to measure changes in brain hemodynamics and oxygenation related to neural electrical activity. Adhering to PRISMA guidelines, results of this search found that generally the majority of studies involving healthy adult subjects observed increased activity in response to odors. Other population types were also observed, such as infants, individuals with autism, attention deficit hyperactivity disorder (ADHD), post-traumatic stress disorder (PTSD), mild cognitive impairment (MCI) and dysosmia. fNIRS coverage heavily favored the prefrontal cortex, temporal and parietal regions. This review finds that odor induced cortical activation is dependent on multiple factors, such as odorant type, gender and population type. This review also finds that there is room for improvement in areas such as participant diversity, use of wearable fNIRS systems, physiological monitoring and multi-distance channels

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.

Are Well Performing Catalysts for the Ring Opening Polymerization of l -Lactide under Mild Laboratory Conditions Suitable for the Industrial Process? the Case of New Highly Active Zn(II) Catalysts

Poly(lactic acid) (PLA) is one of the best candidates as a sustainable plastic material for a circular economy, being biodegradable, bio-based, recyclable, and displaying good thermal and mechanical properties. The industrial production of PLA is mainly based on the ring opening polymerization (ROP) of l-lactide (l-LA) promoted by tin(II) 2-ethylhexanoate [Sn(Oct)2] in a continuous solvent-free process operating at temperatures between 180 and 200 °C, above the melting point of the resulting isotactic polymer. Despite the huge efforts in the research of alternative catalysts based on less toxic metals, resulting in a plethora of highly active catalysts under laboratory mild conditions, very few candidates can compete with Sn(Oct)2 under industrially relevant conditions. We report a family of new Zn(II) complexes, bearing variously substituted monoanionic [N,O-] (imidazole[1,5-a]pyrid-3-yl)phenolate ligands, as catalysts for the ROP of l-LA under both mild (20 °C, solvent) and industrially relevant (190 °C, in the melt, technical grade unpurified monomer, very low catalyst loading) conditions. Interestingly, the best performing catalyst under mild conditions is the worst performing under harsh conditions, and, on the contrary, the less active catalysts under mild conditions compete well with Sn(Oct)2 under industrially relevant conditions. Kinetic and DFT mechanistic investigations shed light on the non-trivial role of the 2-pyridine substituent in the catalytic performances at different temperatures. Preliminary depolymerization tests on commercial PLLA samples suggested that the new catalysts can also be a suitable candidate for the chemical recycling of PLA under mild conditions

Symmetries of the Dirac operators associated with covariantly constant Killing-Yano tensors

The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmetry can be only U(1) and SU(2), specific to (hyper-)Kahler spaces, but arising even in cases when the requirements for these special geometries are not fulfilled. The discrete symmetries are also studied obtaining the discrete groups Z_4 and Q. The briefly presented examples are the Euclidean Taub-NUT space and the Minkowski spacetime.Comment: 27 pages, latex, no figures, final version to be published in Class. Quantum Gravit

Gravitational Waves: Just Plane Symmetry

We present some remarkable properties of the symmetry group for gravitational plane waves. Our main observation is that metrics with plane wave symmetry satisfy every system of generally covariant vacuum field equations except the Einstein equations. The proof uses the homothety admitted by metrics with plane wave symmetry and the scaling behavior of generally covariant field equations. We also discuss a mini-superspace description of spacetimes with plane wave symmetry.Comment: 10 pages, TeX, uses IOP style file

Noether theorem for mu-symmetries

We give a version of Noether theorem adapted to the framework of mu-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of lambda-symmetries, and connects mu-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "mu-conservation law'' actually reduces to a standard one; we also note a relation between mu-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under mu-prolonged variation fields, leading to modified Euler-Lagrange equations. In this setting mu-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples.Comment: 28 pages, to appear in J. Phys.

Towards the theory of integrable hyperbolic equations of third order

The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified Landau--Lifshitz system. The problem of choice of dynamical variables for the hyperbolic equations is discussed.Comment: 22