- CHAOS, TIME AND TEMPORAL NUMBERS

Andrei Vitalyevich Kolesnikov, Svetlana Nikolaevna Sirenko and Georgy Gennadjevich Malinetsky

The concept of time differs significantly in philosophy and the humanities, on the one hand, and mathematics and physics, on the other. Mathematics and physics traditionally focus on reproducible periodic processes for the purpose of predicting their results. Philosophy and the humanities mostly consider time as a historical irreproducible flow of unique events that create history and the unique development scenario. An important scientific issue is to converge of both concepts, which is possible due to the modern views on the deterministic chaos theory. The concepts “dynamic set”, “true”, or “temporal” number should be used to adequately describe historical time. The characteristics of deterministic chaos processes with temporal numbers can be showed on an example of computational models of continuous cellular automata whose rules of transition between states include nonlinearity

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Page No: 4927-4934 - OPTIMAL CONTROL LAWS FOR A PENSION PLAN WITH AND WITHOUT RETURN CLAUSE AND VOLATILITY RISK

Edikan E. Akpanibah and Bright O. Osu

This paper merged together the study of optimal control laws for a pension plan with and without return clause under Heston volatility model. An investment model comprising of members’ monthly contributions, return accumulations with risk free interest to dead members’ families for the case with return clause and investment in one risk free asset and two risky assets is presented. Since the mean variance utility function is time inconsistent, the game theoretic approach is used to establish an optimization problem from the extended Hamilton Jacobi Bellman (HJB) equation. Furthermore, the optimal control laws for the three assets and the efficient frontier are obtained using variable separation method by solving the extended HJB equations. Finally, Numerical simulations were presented to demonstrate the effects of some parameters on the optimal control laws with observations that the optimal control law for risk free asset decreases continuously with time while that of the risky assets increases continuously with time.

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Page No: 4935-4946 - ON EVALUATIONS OF FAST SPEEDS OF PROPAGATION OF GRAVITATIONAL PHENOMENA: A REVIEW

Aleksey Anatolievich Zakharenko

To discuss of evaluations of propagation speeds of gravitational phenomena in continuous media is the main purpose of this review. This work touches the original work by Newton published in 1687, in which the gravitational phenomena propagate with an instant speed. In the second half of 1700s, Le Sage and Laplace have independently evaluated the propagation speeds of the gravitational phenomena to resolve the problem of the instant propagation. Each of them has actually obtained noninstant speeds for the phenomena but many orders faster than the speed of light in a vacuum. Several interesting independent evaluations were carried out in the last and this centuries by Kozyrev, Atsukovsky, Fedulaev, and the others. These evaluations have confirmed that the propagation speeds are almost instant but not instant. In the 1950s, Kozyrev has even created his own nonoptic experimental tools to observe true positions of stars distant from the earth on 10 to 1000 light years. Today some research teams have successfully realized new communication based on the fast speed gravitational phenomena that can be used for the instant interplanetary communication. Also, some researchers have recently proposed fast interplanetary spaceships with gravitational phenomena engines. It is expected that this review can be useful for both researchers-engineers working in the gravitational phenomena research arena and students for educational purposes. This review can also contribute to the successful creation of perfect infrastructure for the instant interplanetary (interstellar and even intergalactic) communication because some developing research directions were discussed

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Page No: 4947-4963 - PLANETARY DIVERGENCE EFFECT AND THE EVOLUTION OF SATURNIAN SYSTEM

Vasily Leonidovich Yanchilin

The paper introduces two new concepts for planetary and satellite systems: the equivalent energy orbit and the equivalent momentum orbit. It is proved that for circular orbits lying in the same plane, the equivalent energy orbit is always located below the equivalent momentum orbit. It is shown that due to energy losses caused by tidal friction, the equivalent energy orbit will gradually drop. Therefore, it is assumed that there is an effect of secular divergence of orbits that acts in planetary and satellite systems. We will term this effect as the Planetary divergence effect. A new hypothesis of the origin of Iapetus, Phoebe, Hyperion, and other external irregular satellites is proposed. The origin of the dichotomy and equatorial ridge of Iapetus, the origin of the dust ring of Phoebe and its retrograde orbit, the origin of Hyperion and its rapid proper rotation are explained. A new hypothesis of the formation of rings of Saturn and other giant planets is proposed. A series of experiments is proposed to comprehensively verify the planetary divergence effect.

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Page No: 4965-4974 - ON SOME COMBINATORIAL RESULTS OF COLLAPSE IN PARTIAL TRANSFORMATION SEMIGROUPS

Mbah, MA, Ndubuisi, RU and Achaku DT

This paper considers the semigroup of partial transformation n P and investigated the elements of collapse given by (eq) . Consequently, the formular for the total number of collapsible elements for (eq) is presented

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