Kinematic fingerprint of core-collapsed globular clusters

Dynamical evolution drives globular clusters toward core collapse, which strongly shapes their internal properties. Diagnostics of core collapse have so far been based on photometry only, namely on the study of the concentration of the density profiles. Here we present a new method to robustly identify core-collapsed clusters based on the study of their stellar kinematics. We introduce the \textit{kinematic concentration} parameter, $c_k$, the ratio between the global and local degree of energy equipartition reached by a cluster, and show through extensive direct $N$-body simulations that clusters approaching core collapse and in the post-core collapse phase are strictly characterized by $c_k>1$. The kinematic concentration provides a suitable diagnostic to identify core-collapsed clusters, independent from any other previous methods based on photometry. We also explore the effects of incomplete radial and stellar mass coverage on the calculation of $c_k$ and find that our method can be applied to state-of-art kinematic datasets.Comment: Accepted for publication in MNRAS Lette

The boundary Riemann solver coming from the real vanishing viscosity approximation

We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems: v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x = \epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx} The conservative case is, in particular, included in the previous formulation. We suppose that the solutions $v^{\epsilon}$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$. Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added Section 3.1.2. Minor changes in other section

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields $$u_t + f(u)_x = 0,$$ then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families $$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N,$$ and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure \mu_{i,\jump} is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\mathrm{jump}}$ is positive

Reducing the linewidth of a diode laser below 30 Hz by stabilization to a reference cavity with finesse above 10^5

An extended cavity diode laser operating in the Littrow configuration emitting near 657 nm is stabilized via its injection current to a reference cavity with a finesse of more than 10^5 and a corresponding resonance linewidth of 14 kHz. The laser linewidth is reduced from a few MHz to a value below 30 Hz. The compact and robust setup appears ideal for a portable optical frequency standard using the Calcium intercombination line.Comment: 8 pages, 4 figures on 3 additional pages, corrected version, submitted to Optics Letter

Quadratic interaction functional for general systems of conservation laws

For the Glimm scheme approximation u_\e to the solution of the system of conservation laws in one space dimension \begin{equation*} u_t + f(u)_x = 0, \qquad u(0,x) = u_0(x) \in \R^n, \end{equation*} with initial data $u_0$ with small total variation, we prove a quadratic (w.r.t. \TV(u_0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux $f$ are made (apart smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely we obtain the following results: \begin{itemize} \item a new analysis of the interaction estimates of simple waves; \item a Lagrangian representation of the derivative of the solution, i.e. a map $\mathtt x(t,w)$ which follows the trajectory of each wave $w$ from its creation to its cancellation; \item the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; \item a new functional $\mathfrak Q$ controlling the variation in speed of the waves w.r.t. time. \end{itemize} This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution $D_{tt} \mathtt x(t,w)$ is a measure with total mass \leq \const \TV(u_0)^2

A Framework to Assess Social Indicators in a Circular Economy Perspective

The thriving circular economy is expected to contribute to all three dimensions of sustainable development: environmental, economic, and social. This paper aims to propose a framework to assess social indicators to support circular business models. To validate the framework, we conducted a case study in a medium-size Italian footwear luxury industry, using the Value Focused Thinking-VFT. This approach was used to define proper social indicators to measure the perception of the company's managers related to the level of incorporation of social dilemmas in the company. We collect data through interviews, documental analysis and direct observation from October/2019 until August/2020 and apply a questionnaire in 2020/2021. The novelty of this paper lies in the proposition of a framework to assess the social indicators in broad categories, capable of covering all supply chains: Corporation, Community; Consumers; Suppliers; Human Rights and Human Resources. Another novelty is related to the analysis of indicators in terms of strategic, tactical, and operational levels, similarly to the idea of a Balanced Scorecard, which was allowed by applying the VFT approach

Overcoming the Main Barriers of Circular Economy Implementation through a New Visualization Tool for Circular Business Models

There is a huge gap between the broad concept of circular economy (CE) and its practical implementation in the industrial sector due to several types of barriers, which shall be led back to the lack of consistent and precise information about resources, products, and processes. Without a proper information flow, it is impossible to quantify circular initiatives, both in comparison with the actual linear situation or with circular alternative opportunities. A proper quantification of circular initiatives allows the assessment of economic, environmental and social benefits and the preventative identification of potential barriers and relative solutions, monitoring the risk associated with circular investments and supporting the decision-making process. This paper describes a new tool to ensure the quantification of circular initiatives and the method to define it. It is a new Circular Business Model (CBM) visualization tool, which overcomes the main limitations of the existing models able to explain CE concepts but not to boost its practical implementation in industry. The new CBM visualization tool can be adopted in every industrial sector to highlight circular opportunities that are still hidden or unexploited or to select the best CE strategy. The proposed CBM visualization tool differs from the previous diagrams in two main characteristics: (i) the possibility to quantify resource flows and important indicators representing energy consumption, environmental and social impact, and (ii) the focus, which is not only on the product, but on the whole system, involving also the process, the company and the entire supply chain. The methodology to adopt and adapt the proposed model to different scales is described in detail. To provide a practical example, the model was qualitatively applied to a generic technical product to highlight its potential in the identification and quantification of circular activities

SBV regularity of Systems of Conservation Laws and Hamilton-Jacobi Equation

We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper