Dynamic behavior of a nonlinear macro-financial system

We investigate dynamic behavior of the macro- financial models governed by a system of three first order differential equation involving interest rate, price exponent and investment demand. Using this mathematical model, all the possible behavior that a model shows in the operation of macro-financial system were examined, such as equilibria, stability and Hopf-bifurcations. We find out the ranges of parameters involved in the system under which the equilibria exist the relationship between the parameters and Hopf-bifurcation. Due to changes in conditions in parameters in this financial system, all the behavior of the model as well as the effects of adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior were discussed by applying Ruth-Hurwitz theorem. Hence, It provides better understanding of the lever function of all types of financial policies.

Dynamic behavior of a nonlinear macro-financial system

We investigate dynamic behavior of the macro- financial models governed by a system of three first order differential equation involving interest rate, price exponent and investment demand. Using this mathematical model, all the possible behavior that a model shows in the operation of macro-financial system were examined, such as equilibria, stability and Hopf-bifurcations. We find out the ranges of parameters involved in the system under which the equilibria exist the relationship between the parameters and Hopf-bifurcation. Due to changes in conditions in parameters in this financial system, all the behavior of the model as well as the effects of adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior were discussed by applying Ruth-Hurwitz theorem. Hence, It provides better understanding of the lever function of all types of financial policies.

Stability analysis and prevention strategies of tobacco smoking model

This research work is related to a tobacco smoking model having a significance class of users of tobacco in the form of snuffing. For this purpose, the formulation of the model containing snuffing class is presented; then the equilibrium points as regards being smoking free and smoking positive are discussed. The Hurwitz theorem is used for finding the local stability of the model and Lyaponov function theory is used for the search of global stability. We use different controls for control of smoking and the Pontryagin maximum principle for characterization of the optimal level. For the solution of the proposed model, a nonstandard finite difference (NSFD) scheme and the Runge–Kutta fourth order method are used. Finally, some numerical results are presented for control and without control systems with the help of MATLAB.

Stability analysis of Beck's column over a fractional-order hereditary foundation

This paper considers the case of Beck's column resting on a hereditary bed of independent springpots. The springpot possesses an intermediate rheological behaviour among linear spring and linear dashpot. It is defined by means of couple ( C β , β ) that characterize the material of the element and is ruled by a Caputo's fractional derivative. In this paper, we investigate the critical load of the column under the action of a follower load by means of a novel complex transform that allows to use the Routh–Hurwitz theorem in the complex half-plane for the stability analysis.

Synchronization behavior of triple-rotor-pendula system in a dual-super-far resonance system

In this paper, a novel design for vibrating screens, called the triple-rotor-pendula system, is presented in a dual-super-far resonance system, which makes it possible to produce wanted vibrations with self-adjustable of the synchronization state between unbalanced rotors. To grasp the synchronization characteristics of the system, the dynamics equation of the triple-rotor-pendula system is primarily derived by adopting Lagrange’s equations. Next, the displacement responses of the system in steady state are obtained by Laplace’s transform method. Meanwhile, considering the average method with revised small parameter, the synchronization implement of the system is ascertained on condition that the absolute values of residual electromagnetism torques between two arbitrary motors are less than or equal to their load torque differences. Then, according to Routh–Hurwitz theorem and generalized Lyapunov equation, the criterion of synchronization stability of the system is obtained. Finally, the simulation computations confirm the results of analytical investigations, which show that the theoretical analysis for the synchronization behavior of the triple-rotor-pendula system is feasible. It can be known that the synchronization state of the system is influenced by mass ratio coefficients, structure parameters, rotating directions, and frequency ratios.

Introduction to Diophantine Approximation. Part II

SummaryIn the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2− a2b1| ≠ 0, has at least one integer solution.

A Riemann–Hurwitz Theorem for the Algebraic Euler Characteristic

We prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.