Envelope Theorems in Economics: Historical Development and Modern Cost-Benefit Applications

This paper reviews some historical development and modern applications of the envelope theorems in economics from a static to a dynamic context. First, we show how the static version of the theorem surfaced in economics, which had eventually lead to the well-known Shephard’s lemma in microeconomics. Second, we present its dynamic version in terms of the classical calculus of variations and optimal control theory via the optimized Hamiltonian function. Third, we show some applications of the theorem for deriving dynamic cost-benefifit rules with special reference to environmental projects involving the green or comprehensive net national product (CNNP). Finally, we illustrate how to extend the cost-benefifit rules to a stochastic economic growth setting.

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.

Global searches of frozen orbits around an oblate Earth-like planet

A frozen orbit is beneficial for observation owing to its stationary apsidal line. The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J2 and limited high-order terms, which cannot meet the stringent demands of all missions. In this study, the gravitational field is expanded to J15 terms and the Hamiltonian canonical form described by the Delaunay variables is used. The zonal harmonic coefficients of the Earth are chosen as the sample. Short-periodic terms are eliminated based on the Hori-Lie transformation. An algorithm is developed to solve all equilibrium points of the Hamiltonian function. A stable frozen orbit with an argument of perigee that equals neither 90° nor 270° is first reported in this paper. The local stability and topology of the equilibrium points are obtained from their eigenvalues. The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods. The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case. The analytical results can be applied to other Earth-like planets and asteroids.

Algebraic Structure and Poisson Integral Method of Snake-Like Robot Systems

The algebraic structure and Poisson's integral of snake-like robot systems are studied. The generalized momentum, Hamiltonian function, generalized Hamilton canonical equations, and their contravariant algebraic forms are obtained for snake-like robot systems. The Lie-admissible algebra structures of the snake-like robot systems are proved and partial Poisson integral methods are applied to the snake-like robot systems. The first integral methods of the snake-like robot systems are given. An example is given to illustrate the results.

Energy and Personality: A Bridge between Physics and Psychology

The objective of this paper is to present a mathematical formalism that states a bridge between physics and psychology, concretely between analytical dynamics and personality theory, in order to open new insights in this theory. In this formalism, energy plays a central role. First, the short-term personality dynamics can be measured by the General Factor of Personality (GFP) response to an arbitrary stimulus. This GFP dynamical response is modeled by a stimulus–response model: an integro-differential equation. The bridge between physics and psychology appears when the stimulus–response model can be formulated as a linear second order differential equation and, subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian equation is derived from a minimum action principle, obtaining the current Lagrangian and Hamiltonian functions. However, the Hamiltonian function is non-conserved energy. Then, some changes lead to a conserved Hamiltonian function: Ermakov–Lewis energy. This energy is presented, as well as the GFP dynamical response that can be derived from it. An application case is also presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol. This experiment provides an ordinal scale for the Ermakov–Lewis energy that predicts the effect of a single dose of alcohol.

Energy and Personality: A Bridge between Physics and Psychology

The objective of this paper is to present a mathematical formalism that states a bridge between Physics and Psychology, concretely between analytical dynamics and personality theory in order to open new insights in this theory. In this formalism energy plays a central role. First, the short-term personality dynamics can be measured by the General Factor of Personality (GFP) response to an arbitrary stimulus. This GFP dynamical response is modelled by a stimulus-response model: an integro-differential equation. The bridge between Physics and Psychology is provided when the stimulus-response model can be formulated as a linear second order differential equation and, subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian equation is derived from a minimum action principle, obtaining the current Lagrangian and Hamiltonian functions. However, the Hamiltonian is a non-conserved energy. Then, some changes provide a conserved Hamiltonian function: the Ermakov-Lewis energy. This energy is presented, as well as the GFP dynamical response that can be derived from it. An application case is presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol. This experiment provides an ordinal scale for the Ermakov-Lewis energies that predicts the effect of a single dose of alcohol.

The Hamiltonian of the f(R) gravity

We derived conjugate momenta variables tensors and the Hamiltonian equation of the source free f(R) gravity from first principles using the Legendre transformation of these conjugate momenta variable tensors, conjugate coordinates variables - fundamental metric tensor and its first ordinary partial derivatives with respect to spacetime coordinates and second ordinary partial derivatives with respect to spacetime coordinates - and the Lagrangian of the f(R) gravity. Interpreting the derived Hamiltonian as the energy of the f(R) gravity we have shown that it vanishes for linear Lagrangians in Ricci scalar curvature without source (e.g. Einstein-Hilbert Lagrangian without matter fields) which is the same result obtained using the stress-energy tensor equation derived from variation of the matter field Lagrangian density. The resulting Hamiltonian equation forbids any model of negative power law in the dependence of the f(R) gravity on Ricci scalar curvature: f(R) = α R^-r, where r and α are positive real numbers, it also forbids any polynomial equation which contains terms with negative powers of the Ricci scalar curvature including a constant term, in which cases the Hamiltonian function in the Ricci scalar and therefore the energy of the f(R) gravity would attains a negative value and would not be bounded from below. The restrictions imposed by the non-negative Hamiltonian have far reaching consequences as the result of applying the f(R) gravity to the study of Black Holes and the Friedmann-Lemaître-Robertson-Walker model in Cosmology.

Analysis of Time-Delay Epidemic Model in Rechargeable Wireless Sensor Networks

With the development of wireless rechargeable sensor networks (WRSNs), many scholars began to attach attention to network security under the spread of viruses. This paper mainly studies a novel low-energy-status-based model SISL (Susceptible, Infected, Susceptible, Low-Energy). The conversion process from low-energy nodes to susceptible nodes is called charging. It is noted that the time delay of the charging process in WRSNs should be considered. However, the charging process and its time delay have not been investigated in traditional epidemic models in WRSNs. Thus, the model SISL is proposed. The basic reproduction number, the disease-free equilibrium point, and the endemic equilibrium point are discussed here. Meanwhile, local stability and global stability of the disease-free equilibrium point and the endemic equilibrium point are analyzed. The addition of the time-delay term needs to be analyzed to determine whether it affects the stability. The intervention treatment strategy under the optimal control is obtained through the establishment of the Hamiltonian function and the application of the Pontryagin principle. Finally, the theoretical results are verified by simulations.

Dynamical Analysis and Optimal Control for a SEIR Model Based on Virus Mutation in WSNs

With the rapid development of science and technology, the application of wireless sensor networks (WSNs) is more and more widely. It has been widely concerned by scholars. Viruses are one of the main threats to WSNs. In this paper, based on the principle of epidemic dynamics, we build a SEIR propagation model with the mutated virus in WSNs, where E nodes are infectious and cannot be repaired to S nodes or R nodes. Subsequently, the basic reproduction number R0, the local stability and global stability of the system are analyzed. The cost function and Hamiltonian function are constructed by taking the repair ratio of infected nodes and the repair ratio of mutated infected nodes as optimization control variables. Based on the Pontryagin maximum principle, an optimal control strategy is designed to effectively control the spread of the virus and minimize the total cost. The simulation results show that the model has a guiding significance to curb the spread of mutated virus in WSNs.

A Within-host Tuberculosis Model Using Optimal Control

In this paper, we studied a mathematical model of tuberculosis with vaccination for the treatment of tuberculosis. We considered an in-host tuberculosis model that described the interaction between Macrophages and Mycobacterium tuberculosis and investigated the effect of vaccination treatments on uninfected macrophages. Optimal control is applied to show the optimal vaccination and effective strategies to control the disease. The optimal control formula is obtained using the Hamiltonian function and Pontryagin's maximum principle. Finally, we perform numerical simulations to support the analytical results. The results suggest that control or vaccination is required if the maximal transmission of infection rate at which macrophages became infected is large. In this paper, we studied a mathematical model of tuberculosis with vaccination for the treatment of tuberculosis.We considered an in-host tuberculosis model that described the interaction between macrophages Macrophages and Mycobacterium tuberculosis and investigated the effect of vaccination treatments on uninfected macrophages. Optimal controlis applied to show the optimal vaccination and effective strategies to control the disease. The optimal control formula isobtained using the Hamiltonian function and Pontryagin's maximum principle. Finally, we perform numerical simulations to support the analytical results.The results suggest thatcontrol or vaccination is required if the maximal transmission of infection rate at which macrophages became infected is large.