A reverse coarea-type inequality in Carnot groups

We prove a coarea-type inequality for a continuously Pansu differentiable function acting between two Carnot groups endowed with homogeneous distances. We assume that the level sets of the function are uniformly lower Ahlfors regular and that the Pansu differential is everywhere surjective.Comment: title changed; various changes of notation; a correction in Theorem 3.2; small changes in the structure; some results extended, typos corrected. The article will appear in Annales de l'Institut Fourie

Area of intrinsic graphs in homogeneous groups

We establish an area formula for computing the spherical measure of an intrinsic graph of any codimension in an arbitrary homogeneous group. Our approach only assumes that the map generating the intrinsic graph is continuously intrinsically differentiable. The important novelty lies in the notion of Jacobian, which is built by the auxiliary Euclidean distance. The introduction of this Jacobian allows the spherical factor to appear in the area formula and enables explicit computations.Comment: 46 page

The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the $k$-th order momenta with respect to the heat kernel

Area formula for regular submanifolds of low codimension in Heisenberg groups

We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is computed with respect to an arbitrary homogeneous distance. Among the arguments of the proof, we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsic differentiable functions.Comment: 27 page

Exploring AuRh Nanoalloys: A Computational Perspective on the Formation and Physical Properties

We studied the formation of AuRh nanoalloys (between 20-150 atoms) in the gas phase by means of Molecular Dynamics (MD) calculations, exploring three possible formation processes: one-by-one growth, coalescence, and nanodroplets annealing. As a general trend, we recover a predominance of Rh@Au core-shell ordering over other chemical configurations. We identify new structural motifs with enhanced thermal stabilities. The physical features of those selected systems were studied at the Density Functional Theory (DFT) level, revealing profound correlations between the nanoalloys morphology and properties. Surprisingly, the arrangement of the inner Rh core seems to play a dominant role on nanoclusters' physical features like the HOMO-LUMO gap and magnetic moment. Strong charge separations are recovered within the nanoalloys suggesting the existence of charge-transfer transitions.The work has been performed under the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the H2020 Programme; in particular, the author gratefully acknowledges the support of UPV/EHU and the computer resources and technical support provided by EPCC. M.V. and S.C. also thank MIUR-FARE for funding under the Grant Plasmochem. R.DA. acknowledges support from the "Grupos Consolidados UPV/EHU del Gobierno Vasco" (Grant No. IT1249-19), and Grant QuEST (Grant No. PID2020-112811GB-I00) funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe" by the European Union. RMJ acknowledges funding by the Engineering and Physical Sciences Research Council (EPSRC) through the Centre for Doctoral Training Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES, Grant No. EP/L015854/1). FB and RMJ thanks the financial support offered by the Royal Society under project number RG120207. We are grateful to the UK Materials and Molecular Modelling Hub for computational resources, partially funded by EPSRC (EP/P020194/1 and EP/T022213/1), our membership of the Materials Chemistry Consortium, funded by EPSRC (EP/R029431). Open Access Funding provided by Universita degli Studi di Milano within the CRUI-CARE Agreement

Formation and relaxation of excited states in solution: A new time dependent polarizable continuum model based on time dependent density functional theory

In this paper a novel approach to study the formation and relaxation of excited states in solution is presented within the integral equation formalism version of the polarizable continuum model. Such an approach uses the excited state relaxed density matrix to correct the time dependent density functional theory excitation energies and it introduces a state-specific solvent response, which can be further generalized within a time dependent formalism. This generalization is based on the use of a complex dielectric permittivity as a function of the frequency, εˆ(ω). The approach is here presented in its theoretical formulation and applied to the various steps involved in the formation and relaxation of electronic excited states in solvated molecules. In particular, vertical excitations (and emissions), as well as time dependent Stokes shift and complete relaxation from vertical excited states back to ground state, can be obtained as different applications of the same theory. Numerical results on two molecular systems are reported to better illustrate the features of the model

Azurin for Biomolecular Electronics: a Reliability Study

The metalloprotein azurin, used in biomolecular electronics, is investigated with respect to its resilience to high electric fields and ambient conditions, which are crucial reliability issues. Concerning the effect of electric fields, two models of different complexity agree indicating an unexpectedly high robustness. Experiments in device-like conditions confirm that no structural modifications occur, according to fluorescence spectra, even after a 40-min exposure to tens of MV/m. Ageing is then investigated experimentally, at ambient conditions and without field, over several days. Only a small conformational rearrangement is observed in the first tens of hours, followed by an equilibrium state

Freezing of gait in Parkinson’s disease patients treated with bilateral subthalamic nucleus deep brain stimulation: A long-term overview

Bilateral subthalamic nucleus deep brain stimulation (STN-DBS) is an effective treatment in advanced Parkinson’s Disease (PD). However, the effects of STN-DBS on freezing of gait (FOG) are still debated, particularly in the long-term follow-up (>/=5-years). The main aim of the current study is to evaluate the long-term effects of STN-DBS on FOG. Twenty STN-DBS treated PD patients were included. Each patient was assessed before surgery through a detailed neurological evaluation, including FOG score, and reevaluated in the long-term (median follow-up: 5-years) in different stimulation and drug conditions. In the long term follow-up, FOG score significantly worsened in the off-stimulation/off-medication condition compared with the preoperative off-medication assessment (z = -1.930; p = 0.05) but not in the on-stimulation/off-medication (z = -0.357; p = 0.721). There was also a significant improvement of FOG at long-term assessment by comparing on-stimulation/off-medication and off-stimulation/off-medication conditions (z = -2.944; p = 0.003). These results highlight the possible beneficial long-term effects of STN-DBS on FOG

Freezing of Gait in Parkinson's Disease Patients Treated with Bilateral Subthalamic Nucleus Deep Brain Stimulation: A Long-Term Overview

Bilateral subthalamic nucleus deep brain stimulation (STN-DBS) is an effective treatment in advanced Parkinson's Disease (PD). However, the effects of STN-DBS on freezing of gait (FOG) are still debated, particularly in the long-term follow-up (≥5-years). The main aim of the current study is to evaluate the long-term effects of STN-DBS on FOG. Twenty STN-DBS treated PD patients were included. Each patient was assessed before surgery through a detailed neurological evaluation, including FOG score, and revaluated in the long-term (median follow-up: 5-years) in different stimulation and drug conditions. In the long term follow-up, FOG score significantly worsened in the off-stimulation/off-medication condition compared with the pre-operative off-medication assessment (z = -1.930; p = 0.05) but not in the on-stimulation/off-medication (z = -0.357; p = 0.721). There was also a significant improvement of FOG at long-term assessment by comparing on-stimulation/off-medication and off-stimulation/off-medication conditions (z = -2.944; p = 0.003). These results highlight the possible beneficial long-term effects of STN-DBS on FOG