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The evolution, as functions of the "ticking time" $\ell =0,1,2,...$, of the solutions of the system of $N$ quadratic recursions \begin{eqnarray*} x_{n}\left( \ell +1\right) =c_{n}+\sum_{m=1}^{N}\left[ C_{nm}x_{m}\left( \ell \right) \right] +\sum_{m=1}^{N}\left\{ d_{nm}\left[ x_{m}\left( \ell \right) \right] ^{2}\right\} +\sum_{m_{1}>m_{2}=1}^{N}\left[ D_{nm_{1}m_{2}}x_{m_{1}}\left( \ell \right) x_{m_{2}}\left( \ell \right) \right] ~,~~~n=1,2,...,N~, && \end{eqnarray*} featuring $N+N^{2}+N^{2}+N\left( N-1\right) N/2=N\left( N+1\right) \left( N+2\right) /2$ ($\ell $-independent) coefficients $c_{n}$, $C_{nm}$, $d_{nm}$ and $D_{nm_{1}m_{2}}$, may be easily ascertained, if these coefficients are given, in terms of $N+N^{2}=N\left( N+1\right) $ a priori arbitrary parameters $a_{n}$ and $b_{nm}$, by $N\left( N+1\right) \left( N+2\right) /2$ explicit formulas provided in this paper. Here $N$ is an arbitrary positive integer.

A kinetic model for the dynamics of collisionless spin neutral particles in a spacetime with torsion is proposed. The fundamental matter field is the kinetic density $f(x,u,s)$ of particles with four-velocity $u$ and four-spin $s$. The stress-energy tensor and the spin current of the particles distribution are defined as suitable integral moments of $f$ in the $(u,s)$ variables. By requiring compatibility with the contracted Bianchi identity in Einstein-Cartan theory, we derive a transport equation on the kinetic density $f$ that generalizes the well-known Vlasov equation for spinless particles. The total number of particles in the new model is not conserved. To restore this important property we assume the existence in spacetime of a second species of particles with the same mass and spin magnitude. The Vlasov equation on the kinetic density $\overline{f}$ of the new particles is derived by requiring that the sum of total numbers of particles of the two species should be conserved., Comment: 26 pages, no figures

In this paper a procedure is described which allows to identify new systems of nonlinear recursions whose solutions are controllable and which may be asymptotically isochronous as functions of the independent variable (considered a ticking time).

In this paper we firstly review how to \textit{explicitly} solve a system of $3$ \textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of $3$ \textit{first-order nonlinear recursions} which also are \textit{explicitly solvable}. These systems might be of interest for practitioners in \textit{applied} sciences: they allow a complete display of their solutions, which may feature interesting behaviors, for instance be \textit{completely periodic} ("isochronous systems", if the independent variable $n=0,1,2,3...$is considered a \textit{ticking time}), or feature this property \textit{only asymptotically} (as\textit{\ }$n\rightarrow \infty $).

In this paper a class of simple, but nonlinear, systems of recursions involving $2$ dependent variables $x_{j}\left( n\right) $ is identified, such that the solutions of their initial-values problems -- with arbitrary initial data $x_{j}\left( 0\right) $ -- may be explicitly obtained.

Bottom-up on-surface synthesis enables the fabrication of carbon nanostructures with atomic precision. Good examples are graphene nanoribbons (GNRs), 1D conjugated polymers, and nanoporous graphenes (NPGs), which are gathering increasing attention for future carbon nanoelectronics. A key step is the ability to manipulate current flow within these nanomaterials. Destructive quantum interference (QI), long studied in the field of single-molecule electronics, has been proposed as the most effective way to achieve such control with molecular-scale precision. However, for practical applications, it is essential that such QI-engineering remains effective near or above room temperature. To assess this important point, here we combine large-scale molecular dynamics simulations and quantum transport calculations and focus our study on NPGs formed as arrays of laterally bonded GNRs. By considering various NPGs with different inter-GNR chemical connections we disentangle the different factors determining electronic transport in these carbon nanomaterials at 300 K. Our findings unequivocally demonstrate that QI survives at room temperature, with thermal vibrations weakly restricting current flow along GNRs while completely blocking transport across GNRs. Our results thus pave the way towards the future realization of QI-engineered carbon nanocircuitry operating at room temperature, which is a fundamental step towards carbon-based nanoelectronics and quantum technologies., Comment: 21 pages, 4 figures

Some simple nonlinear recursions which can be completely managed are identified and the behaviour of all their solutions is ascertained.

In the last years, artificial intelligence (AI) in clinical decision support systems (CDSS) played a key role in harnessing machine learning and deep learning architectures. Despite their promising capabilities, the lack of transparency and explainability of AI models poses significant challenges, particularly in medical contexts where reliability is a mandatory aspect. Achieving transparency without compromising predictive accuracy remains a key challenge. This paper presents a novel method, namely Rad4XCNN, to enhance the predictive power of CNN-derived features with the interpretability inherent in radiomic features. Rad4XCNN diverges from conventional methods based on saliency map, by associating intelligible meaning to CNN-derived features by means of Radiomics, offering new perspectives on explanation methods beyond visualization maps. Using a breast cancer classification task as a case study, we evaluated Rad4XCNN on ultrasound imaging datasets, including an online dataset and two in-house datasets for internal and external validation. Some key results are: i) CNN-derived features guarantee more robust accuracy when compared against ViT-derived and radiomic features; ii) conventional visualization map methods for explanation present several pitfalls; iii) Rad4XCNN does not sacrifice model accuracy for their explainability; iv) Rad4XCNN provides global explanation insights enabling the physician to analyze the model outputs and findings. In addition, we highlight the importance of integrating interpretability into AI models for enhanced trust and adoption in clinical practice, emphasizing how our method can mitigate some concerns related to explainable AI methods.

Ultraviolet nanosecond laser annealing (UV-NLA) proves to be an important technique, particularly when tightly controlled heating and melting are necessary. In the realm of semiconductor technologies, the significance of nanosecond laser annealing (NLA) grows in tandem with the escalating intricacy of integration schemes in nano-scaled devices. Silicon-germanium alloys have been studied for decades for their compatibility with silicon devices. Indeed, they enable the manipulation of properties like strain, carrier mobilities and bandgap. In this framework, they can for instance boost the performances of p-type MOSFETs but also enable near infra-red absorption and emission for applications in photo-detection and photonics. Laser melting on such type of layers, however results, up to now, in the development of extended defects and poor control over layer morphology and homogeneity. In our study, we investigate the laser melting of ~700 nm thick relaxed silicon-germanium samples coated with SiO2 nano-arrays, observing the resulting material to maintain an unaltered lattice. We found the geometrical parameters of the silicon oxide having an impact on the thermal budget samples see, influencing melt threshold, melt depth and germanium distribution.

The properties of liquid Si-Ge binary systems at melting conditions deviate from those expected by the ideal alloy approximation. Particularly, a non-linear dependence of the dielectric functions occurs with the reflectivity of liquid Si-Ge reaching a maximum at 50\% Ge content, being 10\% higher than in pure Si or Ge. Using \textit{ab initio} methodologies, we modelled liquefied Si-Ge alloys, unveiling very high coordination numbers and poor symmetry in the first coordination shell with respect to Si and Ge, related to different bonding properties. We simulated optical functions, quantitatively replicating the aforementioned reflectivity trend and we highlighted a direct relationship between atomic structure and optical properties, indicating that the unusual optics arises from Si-Ge higher local coordination characterized by low symmetry. We forecast further implications for the overall class of these alloys. These findings expand our comprehension of liquefied semiconductors and are essential for implementing controlled laser melting procedures to highly dope these materials for advanced transistors, superconductors, sensors and plasmonic devices.

Controlling ultrafast material transformations with atomic precision is essential for future nanotechnology. Pulsed laser annealing (LA), inducing extremely rapid and localized phase transitions, is a powerful way to achieve this, but it requires careful optimization together with the appropriate system design. We present a multiscale LA computational framework able to simulate atom-by-atom the highly out-of-equilibrium kinetics of a material as it interacts with the laser, including effects of structural disorder. By seamlessly coupling a macroscale continuum solver to a nanoscale super-lattice Kinetic Monte Carlo code, this method overcomes the limits of state-of-the-art continuum-based tools. We exploit it to investigate nontrivial changes in composition, morphology and quality of laser-annealed SiGe alloys. Validations against experiments and phase-field simulations, as well as advanced applications to strained, defected, nanostructured and confined SiGe are presented, highlighting the importance of a multiscale atomistic-continuum approach. Current applicability and potential generalization routes are finally discussed.

We consider a parametric Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and with a reaction which exhibits the combined effects of a superlinear (convex) term and of a negative sublinear term. Using variational tools and critical groups we show that for all small values of the parameter, the problem has at least three nontrivial smooth solutions, two of which are of constant sign (positive and negative).

Public infrastructure procurement is crucial as a prerequisite for public and private investments and for economic and social capital growth. However, low performance in execution severely hinders infrastructure provision and benefits delivery. One of the most sensitive phases in public infrastructure procurement is the design because of the strategic relationship that it potentially creates between procurers and contractors in the execution stage, affecting the costs and the duration of the contract. In this paper, using recent developments in non-parametric frontiers and propensity score matching, we evaluate the performance in the execution of public works in Italy. The analysis provides robust evidence of significant improvement of performance where procurers opt for a design and build contracts, which lead to lower transaction costs, allowing contractors to better accommodate the project in the execution. Our findings bear considerable policy implications.

We consider an anisotropic $(p,2)$-equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques, and critical groups.

Ultraviolet nanosecond laser annealing (LA) is a powerful tool where strongly confined heating and melting are desirable. In semiconductor technologies the importance of LA increases with the increasing complexity of the proposed integration schemes. Optimizing the LA process along with the experimental design is challenging, especially when complex 3D nanostructured systems with various shapes and phases are involved. Within this context, reliable simulations of laser melting are required for optimizing the process parameters while reducing the number of experimental tests. This gives rise to a virtual Design of Experiments (DoE). $Si_{1-x}Ge_{x}$ alloys are nowadays used for their compatibility with silicon devices enabling to engineer properties such as strain, carrier mobilities and bandgap. In this work, the laser melting process of relaxed and strained $Si_{1-x}Ge_{x}$ is simulated with a finite element method / phase field approach. Particularly, we calibrated the dielectric functions of the alloy for its crystalline and liquid phase using experimental data. We highlighted the importance of reproducing the exact reflectivity of the interface between air and the material in its different aggregation states, to correctly mimic the process. We indirectly discovered intriguing features on the optical behavior of melt silicon-germanium.

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer $n_{\ell }$ is even, $n_{\ell +1}=n_{\ell }/2$. The so-called Collatz conjecture states that, starting from any positive integer $N$, the recursion characterized by the continued application of these rules ends up in the cycle $4,$ $2,1$. This conjecture is generally believed to be true (on the basis of extensive numerical checks), but it is as yet unproven. In this paper -- based on the assumption that the Collatz conjecture is indeed true -- we present a quite simple extension of it, which entails the possibility to divide all natural numbers into $3$ disjoint classes, to each of which we conjecture -- on the basis of (not very extensive) numerical checks -- $1/3$ of all natural numbers belong; or, somewhat equivalently, to $2$ disjoint classes, to which we conjecture that respectively $1/3$ and $2/3$ of all natural numbers belong.

In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of first degree divided by the same homogeneous polynomial of second degree. The model features only 4 arbitrary parameters. We also report its isochronous variant featuring 4 nonlinearly-coupled first-order ODEs in 4 dependent variables, featuring 9 arbitrary parameters., Comment: 3 pages

A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints., Comment: 10 pages

Collaborative robots are expected to physically interact with humans in daily living and workplace, including industrial and healthcare settings. A related key enabling technology is tactile sensing, which currently requires addressing the outstanding scientific challenge to simultaneously detect contact location and intensity by means of soft conformable artificial skins adapting over large areas to the complex curved geometries of robot embodiments. In this work, the development of a large-area sensitive soft skin with a curved geometry is presented, allowing for robot total-body coverage through modular patches. The biomimetic skin consists of a soft polymeric matrix, resembling a human forearm, embedded with photonic Fiber Bragg Grating (FBG) transducers, which partially mimics Ruffini mechanoreceptor functionality with diffuse, overlapping receptive fields. A Convolutional Neural Network deep learning algorithm and a multigrid Neuron Integration Process were implemented to decode the FBG sensor outputs for inferring contact force magnitude and localization through the skin surface. Results achieved 35 mN (IQR = 56 mN) and 3.2 mm (IQR = 2.3 mm) median errors, for force and localization predictions, respectively. Demonstrations with an anthropomorphic arm pave the way towards AI-based integrated skins enabling safe human-robot cooperation via machine intelligence., Comment: 6 figures, 4 extended data figures, 2 extended data tables, 39 pages

We study the concept of free will by defining a mathematical model that extends Newton's laws of motion, in such a way that bodies are replaced with agents endowed with free will. In our model the free will of agents is not entirely free, but is bound by the Golden Rule, an ethic found in almost all cultures, which states that one should wish upon others as one wishes upon himself.

The generic monic polynomial of degree N features N a priori arbitrary coefficients $c_m$ and N zeros $z_n$. In this paper we limit consideration to $N = 8$ and $N = 9$. We show that if the $N$ -- a priori arbitrary -- coefficients $c_m$ of these polynomials are appropriately defined -- as it were, a posteriori -- in terms of 6 arbitrary parameters, then the $N$ roots of these polynomials can be explicitly computed in terms of radicals of these 6 parameters. We also report the constraints on the N coefficients $c_m$ implied by the fact that they are so defined in terms of 6 arbitrary parameters; as well as the explicit determination of these 6 parameters in terms of the N coefficients $c_m$., Comment: 5 pages

The initial-values problem of the following nonlinear autonomous recursion of order p , z (s + p) = c product of [z (s + l)]^a_l ; with p an arbitrary positive integer, z (s) the dependent variable (possibly a complex number), s the independent variable (a non negative integer), c an arbitrarily assigned, possibly complex, number, and the p exponents a_l arbitrarily assigned integers (positive, negative or vanishing, so that the right-hand side of the recursion be univalent)|is solvable by algebraic operations, involving the solution of a system of linear algebraic equations (generally explicitly solvable) and of a single polynomial equation of degree p (hence explicitly solvable for p = 1; 2; 3; 4 )., Comment: 7 pages

We study the behavior of solutions for the parametric equation $$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$ under Dirichlet condition, where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain with a $C^2$-boundary $\partial \Omega$, $a_1,a_2 \in L^\infty(\Omega)$ with $a_1(z),a_2(z)>0$ for a.a. $z \in \Omega$, $p,q \in (1,\infty)$ and $\Delta_{p}^{a_1},\Delta_{q}^{a_2}$ are weighted versions of $p$-Laplacian and $q$-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $f(z,x)$ asymptotically as $x \to \pm \infty$ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When $\lambda$ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.

In the ever-increasing energy demand scenario, the development of novel photovoltaic (PV) technologies is considered to be one of the key solutions to fulfil the energy request. In this context, graphene and related two-dimensional (2D) materials (GRMs), including nonlayered 2D materials and 2D perovskites, as well as their hybrid systems, are emerging as promising candidates to drive innovation in PV technologies. The mechanical, thermal, and optoelectronic properties of GRMs can be exploited in different active components of solar cells to design next-generation devices. These components include front (transparent) and back conductive electrodes, charge transporting layers, and interconnecting-recombination layers, as well as photoactive layers. The production and processing of GRMs in the liquid phase, coupled with the ability to on-demand tune their optoelectronic properties exploiting wet-chemical functionalization, enable their effective integration in advanced PV devices through scalable, reliable, and inexpensive printing-coating processes. Herein, we review the progresses in the use of solution-processed 2D materials in organic solar cells, dye-sensitized solar cells, perovskite solar cells, quantum dot solar cells, and organic-inorganic hybrid solar cells, as well as in tandem systems. We first provide a brief introduction on the properties of 2D materials and their production methods by solution-processing routes. Then, we discuss the functionality of 2D materials for electrodes, photoactive layer components-additives, charge transporting layers, and interconnecting layers through figures of merit, which allow the performance of solar cells to be determined and compared with the state-of-the-art values. We finally outline the roadmap for the further exploitation of solution-processed 2D materials to boost the performance of PV devices., Comment: Chemical Society Reviews, 2021, Advance Article

In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data, Comment: 3 Pages. arXiv admin note: substantial text overlap with arXiv:2106.06634

The asymmetric May-Leonard model is a prototypical system of 3 nonlinearly coupled first-order Ordinary Differential Equations with second-degree polynomial right-hand sides. In this short paper we identify a class of special solutions of this system which do not seem to have been previously advertised in spite of their rather elementary character., Comment: 3 pages

This paper discusses the perspective of the H2020 TEACHING project on the next generation of autonomous applications running in a distributed and highly heterogeneous environment comprising both virtual and physical resources spanning the edge-cloud continuum. TEACHING puts forward a human-centred vision leveraging the physiological, emotional, and cognitive state of the users as a driver for the adaptation and optimization of the autonomous applications. It does so by building a distributed, embedded and federated learning system complemented by methods and tools to enforce its dependability, security and privacy preservation. The paper discusses the main concepts of the TEACHING approach and singles out the main AI-related research challenges associated with it. Further, we provide a discussion of the design choices for the TEACHING system to tackle the aforementioned challenges

In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data., Comment: 4 pages

We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a $p$-Laplacian and of a weighted $q$-Laplacian ($q0$, the equation has at least two positive solutions.

A new four-parameters family of constitutive functions for spherically symmetric elastic bodies is introduced which extends the two-parameters class of polytropic fluid models widely used in several applications of fluid mechanics. The four parameters in the polytropic elastic model are the polytropic exponent $\gamma$, the bulk modulus $\kappa$, the shear exponent $\beta$ and the Poisson ratio $\nu\in (-1,1/2]$. The two-parameters class of polytropic fluid models arises as a special case when $\nu=1/2$ and $\beta=\gamma$. In contrast to the standard Lagrangian approach in elasticity theory, the polytropic elastic model in this paper is formulated directly in physical space, i.e., in terms of Eulerian state variables, which is particularly useful for the applications e.g. to astrophysics where the reference state of the bodies of interest (stars, planets, etc.) is not observable. After discussing some general properties of the polytropic elastic model, the steady states and the homologous motion of Newtonian self-gravitating polytropic elastic balls are investigated. It is shown numerically that static balls exist when the parameters $\gamma,\beta$ are contained in a particular region $\mathcal{O}$ of the plane, depending on $\nu$, and proved analytically for $(\gamma,\beta)\in\mathcal{V}$, where $\mathcal{V}\subset\mathcal{O}$ is a disconnected set which also depends on the Poisson ratio $\nu$. Homologous solutions describing continuously collapsing balls are constructed numerically when $\gamma=4/3$. The radius of these solutions shrinks to zero in finite time, causing the formation of a center singularity with infinite density and pressure. Expanding self-gravitating homologous elastic balls are also constructed analytically for some special values of the shear parameter $\beta$., Comment: 40 pages, 5 figures

We study a class of power-law stored energy functions for spherically symmetric elastic bodies that includes well-known material models, such as the Saint Venant-Kirchhoff, Hadamard, Signorini and John models. We identify a finite subclass of these stored energy functions, which we call Lam\'e type, that depend on no more material parameters than the bulk modulus $\kappa>0$ and the Poisson ratio $-1<\nu\leq1/2$. A general theorem proving the existence of static self-gravitating elastic balls for some power-law materials has been given elsewhere. In this paper numerical evidence is provided that some hypotheses in this theorem are necessary, while others are not., Comment: 21 pages, 10 figures

The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the corresponding polynomial can be explicitly computed in terms of radicals of these parameters. We also report the 2 constraints on the 6 coefficients of the polynomial implied by the fact that they are so defined in terms of 5 arbitrary parameters; as well as the explicit determination of these 5 parameters in terms of the 6 coefficients of the sextic polynomial., Comment: 4 pages

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases., Comment: 18 pages

We propose and investigate the intrinsically thinnest transistor concept: a monolayer ballistic heterojunction bipolar transistor based on a lateral heterostructure of transition metal dichalcogenides. The device is intrinsically thinner than a Field Effect Transistor because it does not need a top or bottom gate, since transport is controlled by the electrochemical potential of the base electrode. As typical of bipolar transistors, the collector current undergoes a tenfold increase for each 60 mV increase of the base voltage over several orders of magnitude at room temperature, without sophisticated optimization of the electrostatics. We present a detailed investigation based on self-consistent simulations of electrostatics and quantum transport for both electron and holes of a pnp device using MoS$_2$ for the 10-nm base and WSe$_2$ for emitter and collector. Our three-terminal device simulations confirm the working principle and a large current modulation I$_\text{ON}$/I$_\text{OFF}\sim 10^8$ for $\Delta V_{\rm EB}=0.5$ V. Assuming ballistic transport, we are able to achieve a current gain $\beta\sim$ 10$^4$ over several orders of magnitude of collector current and a cutoff frequency up to the THz range. Exploration of the rich world of bipolar nanoscale device concepts in 2D materials is promising for their potential applications in electronics and optoelectronics., Comment: Accepted for publication on Physical Review Research, 15 pages, 3 figures with subfigures

The explicit solution of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODE s with second-degree polynomial right-hand sides, hence featuring 12 a prior arbitrary (time-independent) coefficients. The solution is explicitly provided if the 12 coefficients are expressed by explicitly provided formulas in terms of 10 a prior arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients is also explicitly solved, but it is found to imply as it were, a posterior that the 12 coefficients must then satisfy 4 algebraic constraints, which are explicitly exhibited.Special sub cases are also identified the general solutions of which are completely periodic with a period independent of the initial data, or are characterized by additional restrictions on the coefficients which identify particularly interesting models., Comment: 17 Pages, Submitted to Mathematical Physics, Analysis and Geometry

The solution $x_n\left(t\right)$, $n=1,2,$ of the \textit{initial-values} problem is reported of the \textit{autonomous} system of $2$ coupled first-order ODEs with \textit{homogeneous cubic polynomial} right-hand sides, \begin{eqnarray} \dot{x}_n = c_{n1} \left(x_1\right)^3 + c_{n2}\left( x_1\right)^2 x_2 + c_{n3} x_1 \left(x_2\right)^2+c_{n4} \left(x_2\right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} when the $8$ (time-independent) coefficients $c_{n\ell}$ are appropriately defined in terms of $7$ \textit{arbitrary} parameters, which then also identify the solution of this model. The inversion of these relations is also investigated, namely how to obtain, in terms of the $8$ coefficients $c_{n\ell},$ the $7$ parameters characterizing the solution of this model; and $2$ \textit{constraints} are \textit{explicitly} identified which, if satisfied by the $8$ parameters $c_{n\ell },$ guarantee the \textit{solvability by algebraic operations} of this dynamical system. Also identified is a related, \textit{appropriately modified}, class of (generally \textit{complex}) systems, reading \begin{eqnarray} \dot{{\tilde{x}}_{n}} = \mathbf{i}\omega \tilde{x}_{n} + c_{n1}\left(\tilde{x}_{1}\right) ^{3}+c_{n2}\left( \tilde{x}_{1}\right) ^2 \tilde{x}_2 + c_{n3}\tilde{x}_1 \left( \tilde{x}_2\right)^2 + c_{n4}\left(\tilde{x}_2 \right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} with $\mathbf{i}\omega $ an \textit{arbitrary imaginary} parameter, which feature the remarkable property to be \textit{isochronous}, namely their \textit{generic} solutions are -- as functions of \textit{real time} -- \textit{completely periodic} with a period which is, for each of these models, a \textit{fixed} \textit{integer multiple} of the basic period $\tilde{T}=2\pi /\left\vert \omega \right\vert$., Comment: 25 pages, To be published in Journal of Mathematical Physics (2021)

We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. Equivalently---by considering the analytic continuation of these systems to complex time---their algebraically solvable character corresponds to the fact that their general solution is either singlevalued or features only a finite number of algebraic branch points as functions of complex time (the independent variable). Thus our results provide a major enlargement of the class of solvable systems beyond those with singlevalued general solution identified by Garnier about 60 years ago. An interesting property of several of these new dynamical systems is the elementary character of their general solution, identifiable as the roots of a polynomial with explicitly obtainable time-dependent coefficients. We also mention that, via a well-known time-dependent change of (dependent and independent) variables featuring the imaginary parameter $% \mathbf{i} \omega $ (with $\omega $ an arbitrary strictly positive real number), autonomous variants can be explicitly exhibited of each of the algebraically solvable models we identify: variants which all feature the remarkable property to be isochronous, i.e. their generic solution is periodic with a period that is a fixed integer multiple of the basic period $T=2\pi/\omega$., Comment: 12 pages, 11 figure

In this paper we investigate the property of engulfing for $H$-convex functions defined on the Heisenberg group ${\mathbb{H}}^n$. Starting from the horizontal sections introduced by Capogna and Maldonado, we consider a new notion of section, called ${\mathbb{H}}^n$-section, as well as a new condition of engulfing associated to the ${\mathbb{H}}^n$-sections, for an $H$-convex function defined in ${\mathbb{H}}^n.$ These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the ${\mathbb{H}}^n$-sections, with their engulfing property, will lead to the definition of a pseudo-metric in ${\mathbb{H}}^n$ in a way similar to Aimar, Forzani and Toledano in the Euclidean case. A key role is played by the property of round $H$-sections for an $H$-convex function, and by its connection with the engulfing properties.

In this note we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

Multiple Sclerosis (MS) is a chronic and potentially highly disabling disease that can cause permanent damage and deterioration of the central nervous system. In Europe it is the leading cause of non-traumatic disabilities in young adults, since more than 700,000 EU people suffer from MS. Although recent studies on MS pathophysiology have been provided, MS remains a challenging disease. In this context, thanks to recent advances in software and hardware technologies, computational models and computer simulations are becoming appealing research tools to support scientists in the study of such disease. Thus, motivated by this consideration we propose in this paper a new model to study the evolution of MS in silico, and the effects of the administration of Daclizumab drug, taking into account also spatiality and temporality of the involved phenomena. Moreover, we show how the intrinsic symmetries of the system can be exploited to drastically reduce the complexity of its analysis., Comment: Submitted to CIBB 2019 post proceedings - LNCS

The use of graphene-based electrodes is burgeoning in a wide range of applications, including solar cells, light emitting diodes, touch screens, field-effect transistors, photodetectors, sensors and energy storage systems. The success of such electrodes strongly depends on the implementation of effective production and processing methods for graphene. In this work, we take advantage of two different graphene production methods to design an advanced, conductive oxide- and platinum-free, graphene-based counter electrode for dye-sensitized solar cells (DSSCs). In particular, we exploit the combination of a graphene film, produced by chemical vapor deposition (CVD) (CVD-graphene), with few-layer graphene (FLG) flakes, produced by liquid phase exfoliation. The CVD-graphene is used as charge collector, while the FLG flakes, deposited atop by spray coating, act as catalyst for the reduction of the electrolyte redox couple (i.e., I3-/I-- and Co+2/+3). The as-produced counter electrodes are tested in both I3-/I-- and Co+2/+3-based semitransparent DSSCs, showing power conversion efficiencies of 2.1% and 5.09%, respectively, under 1 SUN illumination. At 0.1 SUN, Co+2/+3-based DSSCs achieve a power conversion efficiency as high as 6.87%. Our results demonstrate that the electrical, optical, chemical and catalytic properties of graphene-based dual films, designed by combining CVD-graphene and FLG flakes, are effective alternatives to FTO/Pt counter electrodes for DSSCs for both outdoor and indoor applications.

We propose an iterative algorithm to interpolate graph signals from only a partial set of samples. Our method is derived from the well known Papoulis-Gerchberg algorithm by considering the optimal value of a constant involved in the iteration step. Compared with existing graph signal reconstruction algorithms, the proposed method achieves similar or better performance both in terms of convergence rate and computational efficiency., Comment: 8 pages, 3 figures

Bottom-up prepared carbon nanostructures appear as promising platforms for future carbon-based nanoelectronics, due to their atomically precise and versatile structure. An important breakthrough is the recent preparation of nanoporous graphene (NPG) as an ordered covalent array of graphene nanoribbons (GNRs). Within NPG, the GNRs may be thought of as 1D electronic nanochannels through which electrons preferentially move, highlighting NPG's potential for carbon nanocircuitry. However, the {\pi}-conjugated bonds bridging the GNRs give rise to electronic cross-talk between the individual 1D channels, leading to spatially dispersing electronic currents. Here, we propose a chemical design of the bridges resulting in destructive quantum interference, which blocks the cross-talk between GNRs in NPG, electronically isolating them. Our multiscale calculations reveal that injected currents can remain confined within a single, 0.7 nm wide, GNR channel for distances as long as 100 nm. The concepts developed in this work thus provide an important ingredient for the quantum design of future carbon nanocircuitry.

The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials., Comment: 30 pages, 2 figures. The order of presentation of the results and the notation have been changed considerably to improve the reading flow of the article. Some assumptions and theorems have been reformulated in a more clear way and several new remarks have been added

We describe a method and its implementation for calculating electronic structure and electron transport without approximating the structure using periodic super-cells. This effectively removes spurious periodic images and interference effects. Our method is based on already established methods readily available in the non-equilibrium Green function formalism and allows for non-equilibrium transport. We present examples of a N defect in graphene, finite voltage bias transport in a point-contact to graphene, and a graphene-nanoribbon junction. This method is less costly, in terms of CPU-hours, than the super-cell approximation., Comment: 8 pages, 7 figures

The evolution equations mentioned in the title of this paper read as follows: x~n = P(n)(x1; x2) , n = 1, 2 , where l is the "discrete-time" independent variable taking integer values (l =0, 1, 2, ...), xn = xn (l) are the 2 dependent variables, x~n = xn (l + 1), and the 2 functions P(n)(x1, x2), n = 1, 2, are 2 polynomials in the 2 dependent variables x1 (l) and x2 (l). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t in which case xn = xn (t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1904.00496

In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: x_n = P(n) (x1, x2) , n = 1, 2 , with P(n) (x1, x2) specific polynomials of relatively low degree in the 2 dependent variables x1 = x1 (t) and x2 = x2 (t) . These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations., Comment: 11 pages, Sabmitted 27 August 2018, to be published as a chapter in a collective book to celebrate the 65th birthdate of Emma Previato (in press)

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,x~n = F^(n)(x1; x2) , n = 1, 2 , which are algebraically solvable. Here l is the "discrete-time" independent variable taking integer values (l = 0, 1, 2,...), xn = xn (l) are 2 dependent variables, and x~n = xn (l + 1) are the corresponding 2 updated variables. In a previous paper the 2 functions F^(n)(x1, x2), n = 1, 2 were defined as follows: F^(n)(x1; x2) = P2 (xn, xn+1), n = 1,2 mod[2]; with P2 (x1; x2) a specific second-degree homogeneous polynomials in the 2 (indistinguishable!) dependent variables x1 (`) and x2 (l). In the present paper we further clarify some aspects of that model and we present its extension to the case when F(n)(x1, x2) = Q^(n)_k(x1, x2), n = 1, 2 mod[2], with Q(n)_k(x1; x2) a specific homogeneous function of arbitrary (integer ) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x1 (l) and x2 (l)., Comment: 13 pages, submitted 13 March 2019

We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of 2 first-order ODEs with polynomial right-hand sides, x_ n= P(n)(x1, x2) , n = 1, 2 , satisfied by the 2 (possibly complex ) dependent variables xn = xn (t). Here P(n)(x1, x2) indicates some specific polynomial. These examples are analogous, but different, from those previously reported., Comment: 9 pages, submitted 29 January 2019

The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; and subsequently significant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. In this paper we introduce an approach suitable to deal with the most general case, i. e. that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. We then focus on the special case of a polynomial of degree 4 featuring only 2 different zeros and, by using a recently introduced additional twist of this approach, we thereby identify many new classes of solvable dynamical systems., Comment: 36 pages, Submitted 11 December, 2018