On the derivation of a nonlinear generalized Langevin equation

We recast the Zwanzig's derivation of a non linear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework showing that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to be able performing consistently the continuum limit. As shown by Zwanzig, the non linear feature of the resulting GLE is due to the non linear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig's Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the non linear GLE.

Mathematical Model Applied to Green Building Concept for Sustainable Cities Under Climate Change

Recently the effect of greenhouse gases (GHGs) is worldwide terrified anxiety to the public and scholars. Even this global problem is one of the great issues that continuously makes worrying the governments and environmentalists, but its solution findings are not out of the image at all. In this study, we have proposed and analysed a mathematical model for the solvable management of GHGs by sowing the seeds of green building dynamic systems. Moreover, in the model, the human community is used to enhance the production power of individuals of green buildings by absorbing the GHGs. The model is analysed by stability analysis at the equilibrium points: trivial and global equilibrium, and also by convincing the stability and instability of the system of equations. The behaviour of the propound model has been developed by numerical simulations which shows the rate of the fruitfulness of GHG components.

Local Turnpike Analysis Using Local Dissipativity for Discrete Time Discounted Optimal Control

Recent results in the literature have provided connections between the so-called turnpike property, near optimality of closed-loop solutions, and strict dissipativity. Motivated by applications in economics, optimal control problems with discounted stage cost are of great interest. In contrast to non-discounted optimal control problems, it is more likely that several asymptotically stable optimal equilibria coexist. Due to the discounting and transition cost from a local to the global equilibrium, it may be more favourable staying in a local equilibrium than moving to the global—cheaper—equilibrium. In the literature, strict dissipativity was shown to provide criteria for global asymptotic stability of optimal equilibria and turnpike behaviour. In this paper, we propose a local notion of discounted strict dissipativity and a local turnpike property, both depending on the discount factor. Using these concepts, we investigate the local behaviour of (near-)optimal trajectories and develop conditions on the discount factor to ensure convergence to a local asymptotically stable optimal equilibrium.

Combining Analytical Models and Mesh Morphing Based Optimization Techniques for the Design of Flying Multihulls Appendages

The fluid dynamic design of hydrofoils involves most of the typical difficulties of aeronautical wings design with additional complexities related to the design of a device operating in a multiphase environment. For this reason, “high fidelity” analysis solvers should be, in general, adopted also in the preliminary design phase. In the case of modern fast foiling sailing yachts, the appendages accomplish both the task of lifting up the boat and to make possible upwind sailing by contributing balance to the sail side force and the heeling moment. Furthermore, their operative design conditions derive from the global equilibrium of forces and moments acting on the system which might vary in a very wide range of values. The result is a design problem defined by a large number of variables operating in a wide design space. In this scenario, the device performing in all conditions has to be identified as a trade-off among several conflicting requirements. One of the most efficient approaches to such a design challenge is to combine multi-objective optimization strategies with experienced aerodynamic design. This paper presents a numerical optimization procedure suitable for foiling multihulls. As a proof of concept, it reports, as an application, the foils design of an A-Class catamaran. The key point of the method is the combination of opportunely developed analytical models of the hull forces with high fidelity multiphase analyses in both upwind and downwind sailing conditions. The analytical formulations were tuned against a database of multiphase analyses of a reference demihull at several attitudes and displacements. An aspect that significantly contributes to both efficiency and robustness of the method is the approach adopted to the geometric parametrization of the foils which was implemented by a mesh morphing technique based on Radial Basis Functions.

On global determinacy of New Keynesian models

New Keynesian models assume that inflation rate and output level are endogenous variables. However, given that firms are price setters and suppliers in the models, it is more reasonable to assume that, absent equilibrium coordination (or tatonnement) issues usually abstracted away, both variables actually are state variables determined by expectations in the past. This secures global equilibrium determinacy and a previously unavailable account of inflation rate for New Keynesian models. Furthermore, the principle of effective demand is implemented via the expectation channel.

The COVID-19 Storm

A brief overview is presented of the global experience with the Novel Coronavirus 19 pandemic and disease (COVID-19) that produced an unanticipated and unprecedented storm in the global equilibrium. The pandemic is described as an external and internal storm, with resultant damage to global and personal health and lifestyles. The mechanisms and implications of the resulting crisis are also touched upon in brief.

Using free association networks to extract characteristic patterns of affect dynamics

The dynamics of human affect in day-to-day life are an intrinsic part of human behaviour. Yet, it is difficult to observe and objectively measure how affect evolves over time with sufficient resolution. Here, we suggest an approach that combines free association networks with affect mapping, to gain insight into basic patterns of affect dynamics. This approach exploits the established connection in the literature between association networks and behaviour. Using extant rich data, we find consistent patterns of the dynamics of the valence and arousal dimensions of affect. First, we find that the individuals represented by the data tend to feel a constant pull towards an affect-neutral global equilibrium point in the valence–arousal space. The farther the affect is from that point, the stronger the pull. We find that the drift of affect exhibits high inertia, i.e. is slow-changing, but with occasional discontinuous jumps of valence. We further find that, under certain conditions, another metastable equilibrium point emerges on the network, but one which represents a much more negative and agitated state of affect. Finally, we demonstrate how the affect-coded association network can be used to identify useful or harmful trajectories of associative thoughts that otherwise are hard to extract.

Variance of Atomic Coordinates as a Dynamical Metric to Distinguish Proteins and Protein-Protein Interactions in Molecular Dynamics Simulations

<b><i>In this manuscript, </i>we introduce the Cumulative Variance of Coordinate Fluctuations (CVCF) along atomistic MD trajectories, as a dynamical metric to examine protein dynamics and sampling convergence in MD simulations.</b> Using model 1D and 2D PES, we first show that CVCF, which traces over the fluctuations of protein atoms as a function of sampling coordinate (time in MD simulations), captures both local and global equilibria to distinguish the underlying PES of proteins. For both model PES and protein trajectories, we compare the information content present in CVCF traces with that obtained using other measures proposed in literature to reveal conditions under which a consistent interpretation of data can be obtained. <b>Importantly, we show that independent of convergence to either local or global equilibrium, the values and features of protein CVCF can provide a comparative assessment of the ruggedness and curvature of the underlying PES sampled by proteins along MD trajectories.</b> Trends in CVCF therefore enable us to compare features of the PES across multiple protein systems using MD simulations. We demonstrate some of the attractive features of a CVCF based analysis on multi-microsecond (ms) MD trajectories of structurally homologous ubiquitin family proteins which present a particularly striking example in nature wherein sequence changes and complexation which do not lead to prominent structural changes bring about dramatic functional consequences.