The paper aims to compare linear time-invariant 1" and 2nd order state equation models (SMx, x = 1, 2 ) of the Global Carbon Cycle that have been estimated from global data, with mid-complexity models (MMx, x = 1, 2, 3, 4) like MAGICC and FAIR. One of the reasons is the different airborne carbon dioxide (CD) prediction, even driven by similar bell-shaped CD emission profiles like SSP245: delayed and amplitude modulated emission that of SMx, sigmoidal that of MMx. To the purpose, the LTI dynamics of MMx has been fitted to a common LTI irrational dynamics describing the vertical carbon diffusion from air to land and ocean. The fitting has proven feasible and accurate, thus providing a standard set of state equations, which can be mechanized to produce centennial predictions under CD emission profiles. Preliminary predictions only differ in amplitude, but not in shape, depending on whether diffusion-fitted LTI dynamics have been obtained either by linearization (MM4) or by extracting the LTI dynamics (MM3). A brief discussion is provided. Copyright (c) 2025 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

The paper has been suggested by the following observations: (1) the atmospheric growth rate of carbon dioxide concentration is smaller than that ascribed to the emission of fossil-fuel combustion and (2) the fossil-fuel reserves are finite. The first observation leads to a simple dynamic model, based on the balance of CO2 land/ocean absorption and anthropogenic emissions, only limited by the depletion of fossil-fuel reserves, in a business-as-usual scenario. The second observation suggests of projecting the past CO2 emissions to the future, by constraining emissions to the limit of reserve availability. Similar projections are available in the literature, but either driven by heuristics or by complex simulation packages. The paper provides a simple and formal method only driven by historical data, their uncertainty and simple models. The method aims to provide CO2 concentration projections, which being constrained by fossil-fuel finite reserve may be in principle employed as bounds to forecasting exercises. The time–invariant dynamics of the land/ocean absorption is the simplification of a more complex set of equations describing carbon dioxide exchange between different reservoirs. Contribution of other greenhouse gases like methane and nitrous oxide has been neglected, since their emissions cannot be projected with the paper methodology. Comparison with recent profiles of the Intergovernmental Panel on Climate Change (IPCC) confirms that the finite-reserve projections of the fossil fuel emissions is close to those of a moderate Shared Socioeconomic Scenario (SSP) like SSP2-4.5—a result in agreement with other authors—but also reveals the limits of the simplified model, when extending the tuned dynamics of the recent mean CO2 exchanges to long-term future. The limits derive from linearity, time invariance, and aggregation assumptions, which allow a more complex model of CO2 exchanges to be simplified and tuned on experimental data.

A significant research activity is ongoing about present and future stability of the Earth's mean global carbon cycle, which plays a key role in climate dynamics, and is perturbed by the increasing carbon emission of the fossil-fuel combustion. The paper aims to provide an answer through a second-order state equation, which is tuned on global data of surface temperature, airborne carbon dioxide (CD) concentration, solar radiation and anthropogenic carbon emissions. The state variables are temperature and airborne CD perturbations from pre-industrial equilibriums. The equilibriums, taken as unknown, are expected to approach historical estimates, thus providing a tuning validation criterion. Since state variables are linked in a closed loop, carbon cycle stability can be assessed through loop stability criteria. The state equations, being linear time invariant, neglect time variability and nonlinearity. They are revealed by a rolling time regression, whose time interval shrinks from industrial era to recent decades, when data become more accurate. The paper concludes with preliminary predictions tuned on recent data. Comparison with predictions of the well-known MAGICC model show discrepancies that are briefly discussed. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

The paper has been suggested by a pair of observations: 1) the atmospheric growth rate of carbon dioxide is smaller than that ascribed to the emission by fossil fuel combustion; 2) the fossil-fuel reserves are finite. The first observation leads to a simple dynamic model, based on the balance between the land-ocean absorption and the anthropogenic emissions of CO2, only limited by the depletion of fossil-fuel reserves. The second observation suggests of projecting the historical CO2 emissions in the future, by constraining them to the limit of reserve availability. Similar projections are available in the literature, but either driven by heuristics or by complex simulation packages. Here we provide a transparent and formal method only driven by historical data, their uncertainty and simple models. The method is proven capable of providing emission and concentration projections, which being constrained by finite reserves, may be taken as realistic bounds to forecasting exercises. The dynamics of the land-ocean absorption is proved by simplifying a more complex set of equations describing the CO2 exchange between Earth's reservoirs. The contribution of other greenhouse gases like methane and nitrous oxide is neglected, as their emissions cannot be projected with the paper method. Notwithstanding this limitation, the paper results demonstrate that some of the IPCC projections are overestimated if compared to fossil-fuel physical limits, in agreement with other authors.

The chapter extends the chemical kinetics of Chapter 3 to more complex chemical reactions, partly discovered in the second half of the 20th century, being of significance for not only chemistry but also general science and dynamic systems. This fact suggests that the treatment employs methods and concepts of dynamic system theory as partly done in Chapter 3. In particular, an extensive use of linear state equations, their state matrix, and the relevant eigenvalues will be done in order to assess stability properties of the chemical kinetics under study. The Chapter starts from the well-known class of Michaelis–Menten reactions in biochemistry. The second reaction is the striking iodine clock, in which the production of triiodide changes the solution color into dark blue. The second part of the chapter is devoted to oscillating chemical reactions (Briggs-Rauscher and Belousov-Zhabotinsky), whose underlying mechanism is that of the classical second-order Lotka–Volterra equations, which are briefly recalled.

The mathematical description of the undulatory behavior of electron orbitals in atoms is shortly reviewed. Orbitals are formulated as spatial wave functions, mathematical solutions of the Schroedinger equation, which are eigenfunctions of spatial operators describing energy and angular momentum of the atom's electron orbit. On this basis, wave and radial functions of the hydrogen-like atom are described and plotted in a simplified manner using Matlab® surf and fsurf functions. A simplified representation of hybrid orbitals is also discussed and plotted. Some elements of wave functions theory and the Schroedinger equation are recalled in Appendix F (Chapter 16).

With the ability to detect low-frequency gravitational waves (GWs), space-borne detectors will play an important role in exploring the universe in the future. The TianQin project proposed in China participates in this challenge by aiming to detect millihertz GWs. The TianQin GW detector consists of three spacecraft forming a triangular constellation. Each spacecraft carries a pair of test masses (TM) which, inside shielding cages, are left to free fall along the local geodesic. In this way, a pair of TMs on two separate spacecraft can be each other aligned to become the gravitational references of the GW detector, made by an inter-satellite laser interferometer. Each free-falling TM is limited by parasitic forces to be appropriately bounded. The relevant acceleration bound has been fixed to 10-15 m/s(2)/ Hz within the TianQin measurement bandwidth (MBW) ranging from 0.1 mHz to 0.1 Hz. In turn, TM to cage fluctuations must be kept below 4nm/Hz to limit the stiffness coupling with the spacecraft displacement caused by non-gravitational disturbances, such as solar radiations and the thruster noise. Suppression of such disturbances calls for a challenging drag-free control technology, since the center of mass (CoM) of a single spacecraft cannot track the separated CoMs of two TMs simultaneously, and consequently tracking must be limited to three degrees of freedom (DoF): two non-orthogonal sensitive axes (one for each TM) and the perpendicular direction to their plane. Control design and simulated tests of this paper will be restricted to the drag-free control along the sensitive axes. The remaining nine DoFs of the two TMs (position and attitude) are controlled by electrostatic suspensions, not to be treated here, but accounted for in the simulated trials. The design of the two-DoF drag-free control relies on a model-based control methodology, the Embedded Model Control (EMC), capable of predicting and suppressing unknown disturbances within the required bandwidth and of decoupling the TM dynamics along the non-orthogonal sensitive axes. The paper starts with the nonlinear model of the TM to cage dynamics, followed by the relevant EMC design, restricted to the sensitive axes. Numerical simulations are employed to validate closed-loop performance and robust stability. Simulated results show that the residual TM to cage fluctuations can be kept below 3nm/root Hz, which leaves a margin within the required spectral bound. The EMC methodology discussed in the paper can provide a reference for future developments and implementations.

Seawater is not a simple reservoir of different dissolved salts, like sodium chloride, but is capable of reacting with different substances like the atmosphere's carbon dioxide, by modifying its concentration and buffering the anthropogenic increase. This is one of today most debated topics. The only quantitative way to tackle the issue is by solving simultaneously all the equations pertaining to the chemical equilibria in seawater. This is discussed, explained, and calculated by the simple use of Matlab® scripts. As with other scripts of this book, the reader can vary the relevant parameters (temperature, hydrostatic pressure, salinity, and others) and discover what happens. The variety of simulations is really amazing, but only some of them are discussed.

Electric interaction between molecules is responsible for the phase transformation of gasses to liquids. A number of equations can be used for describing the behavior of real gases, in a wide range of temperatures and pressures, which are graphically displayed by Matlab® scripts. A gas of choice can be used, as long as its van der Waals parameters are known.

The paper has been suggested by two observations: 1) the atmospheric CO$_2$ growth rate is smaller than that ascribed to the emission of fossil fuels combustion, 2) the fossil fuel reserves are finite. The first observation has lead the way to a simple kinetic mode, based on the balance of 1) land/ocean CO$_2$ absorption and 2) CO$_2$ anthropogenic emission limited solely by depletion of the present day fossil-fuel reserves, in a business-as-usual scenario. The second observation has suggested to extrapolate past CO$_2$ emissions by fossil fuel combustion in the future years up to 2200 CE, by constraining emissions to the physical limits of reserves availability. The Meixner curve (hyperbolic secant distribution) has been used to model the pathway of resource exploitation for the three main classes of fossil fuels, crude oil, natural gas and coal. The kinetic model, driven by the extrapolated emissions, has been employed to project the CO$_2$ atmospheric concentration due to fossil fuel combustion close to the zero-reserve epoch. The result is just the output of simple models tuned on well-known experimental data. Error analysis of literature data provides the method robustness and the relevant uncertainty band. Contribution of other greenhouse gases like methane and nitrous oxide has been neglected, since their emissions cannot be projected with the paper methodology (they do not derive from fossil reserves). Notwithstanding this limitation, paper results clearly demonstrate that some of the IPCC projections of the CO$_2$ concentration are largely overestimated if compared to the physical limits of fossil fuel exploitation.

In general chemistry, this is a relevant topic. Many Matlab® functions, among them fzero, enable in a few program lines to solve even complex equilibria of this kind. Examples range from monoprotic acids to titratrion of polyprotic acids. Though a few working examples are given, the reader will be able to insert other acid/base equilibria in solution, even admixtures of acids and bases. An insight into coupled acid–base and precipitation reaction is given, as an example for the carbonatic acid–base equilibria involving CaCO3 and Mg(OH)2 precipitation.

The algebra underlying the balancing of chemical reactions is highlighted. In such a way, a generic and straightforward solution is devised for simple acid–base or complex redox reactions. This could help solve troublesome issues, being the correct mass balance of utmost relevance in general and educational chemistry. The foundations of the algebraic algorithm employed, based on the nullspace of a matrix and the integer constraint, are reviewed in Appendix A (Chapter 11). Only some examples of reactions are given, while the reader may exploit other as well among the many possible.

Dealing with thermodynamics of even simple equilibrium reactions in a varying temperature/pressure environment requires a precise approach. Matlab® functions enable to read excel files with the parameters needed to calculate entropy, enthalpy, and free energy and therefrom the equilibrium parameters for some reactions. By exploiting the indicated thermodynamic database, other reactions will be tackled, among the many of interest in environmental issues.

The study of the rate of a chemical reactions in both textbooks and advanced applications requires to deal with linear and nonlinear differential equations. Their temporal solution, even for simple applications, is enabled by Matlab® ODE solvers and Matlab/Simulink block diagrams. Extensive use of 2D graphical plot will be done. The treatment agrees with the methods of dynamic systems, whose foundations are recalled in Appendix B (Chapter 12). Two case studies, NO oxidation and ozone decomposition, can be extended to many other known reactions.

The main goal of the paper is to test the Embedded Model Control (EMC) design and implementation on a typical underactuated apparatus, like the Furuta pendulum, by comparing experimental results with a Linear Quadratic Regulator (LQR). EMC can be considered as a disturbance rejection control strategy, since the state predictor is extended to explicitly include disturbance dynamics, in charge of predicting the uncertainty to be rejected by control law. Essential in EMC design is the separation between controllable and not controllable dynamics, a task which allows us to find the controllable channel of underactuated systems from the low-dimensional command to the whole system degrees of freedom (DF). Pursuing this objective, a rather generic method is shown, which is applicable to other underactuated systems. The result is a very simple controllable dynamics from the single pendulum command to pendulum DF arranged in a single series of controllable integrators. The neglected feedback channels, including the unstable gravity feedback, are treated as unknown thus posing a challenge to disturbance prediction and closed loop stability. Typical in EMC, closed loop eigenvalues are chosen to guarantee stability, a pre-requisite to performance. Experimental results point out effectiveness and advantage, with respect to LQR, of design and implementation under adverse conditions, due to a disturbance pulse, in which command saturates.

The paper was suggested by a brief note of the second author about the application of the Hubbert curve to predict decay of resource exploitation. A further suggestion came from the interpretation of the Hubbert curve in terms of a specific Lotka Volterra (LV) equation. The link with population dynamics was obvious as logistic function and LV equation were proposed within the demography science field. Mathematical population dynamics has a history of about two centuries. The first principle and model of population dynamics can be regarded the exponential law of Malthus. In the XIX century, the Malthusian demographic model was first refined to include mortality rate by Gompertz. In the early XIX century the model was further refined by Verhulst by introducing the standard logistic function. The previous models only concern the population of a single species. In the early XX century, the American demographer Lotka and the Italian mathematician Volterra proposed a pair of state equations which describe the population dynamics of two competing species, the predator and the prey. This paper is concerned with the single and two-species fundamental equations: the logistic and LV equation. The paper starts with the generalized logistic equation whose free response is derived together with equilibrium points and stability properties. The parameter estimation of the logistic function is applied to the raw data of the US crude oil production. The paper proceeds with the Lotka Volterra equation of the competition between two species, with the goal of applying it to resource exploitation. At the end, a limiting version of the LV equation is studied since it describes a competition model between the production rate of exploited resources and the relevant capital stock employed in the exploitation.

TianQin is a planned space-based gravitational wave (GW) observatory consisting of three Earth-orbiting satellites with an orbital radius of about $10^5 \, {\rm km}$. The satellites will form an equilateral triangle constellation the plane of which is nearly perpendicular to the ecliptic plane. TianQin aims to detect GWs between $10^{-4} \, {\rm Hz}$ and $1 \, {\rm Hz}$ that can be generated by a wide variety of important astrophysical and cosmological sources, including the inspiral of Galactic ultra-compact binaries, the inspiral of stellar-mass black hole binaries, extreme mass ratio inspirals, the merger of massive black hole binaries, and possibly the energetic processes in the very early universe and exotic sources such as cosmic strings. In order to start science operations around 2035, a roadmap called the 0123 plan is being used to bring the key technologies of TianQin to maturity, supported by the construction of a series of research facilities on the ground. Two major projects of the 0123 plan are being carried out. In this process, the team has created a new-generation $17 \, {\rm cm}$ single-body hollow corner-cube retro-reflector which was launched with the QueQiao satellite on 21 May 2018; a new laser-ranging station equipped with a $1.2 \, {\rm m}$ telescope has been constructed and the station has successfully ranged to all five retro-reflectors on the Moon; and the TianQin-1 experimental satellite was launched on 20 December 2019—the first-round result shows that the satellite has exceeded all of its mission requirements.