A probabilistic solution for the Syracuse conjecture

We prove the veracity of the Syracuse conjecture by establishingthat starting from an arbitrary positive integer diffrent of 1 and 4, theSyracuse process will never comeback to any positive integer reachedbefore and then we conclude by using a probabilistic approach.Classification : MSC: 11A25

SOLAS 2009 STABILITY REQUIREMENTS: IMPLEMENTATION

January 2009 saw the introduction of substantial changes to SOLAS, commonly referred to as SOLAS 2009. Not only have significant parts of Chapter II-1 completely changed, but so have the methodologies for assessing survivability of certain ship types. This paper provides an overview of some of the main topics and how Lloyd’s Register is adapting to provide necessary industry solutions and support, immediately and into the future. It provides an insight into the probabilistic requirements, our approval processes, developments and our participation in defining industry standards. It is evident in this paper that the discussions predominantly revolve around passenger ships. This is due to their complexity and the conflict between the new regulations for survivability assessment moving from a restrained deterministic requirement to a risk-based probabilistic solution. It also highlights real issues over the difficulties of implementing this methodology. This conflict in overall design is less pronounced for dry cargo ships, which did not have to comply with a general damage stability standard until 1992 when the probabilistic concept was introduced for dry cargo ships only. Under SOLAS 2009, a modified requirement has been implemented. However, the fundamental issues remain the same.

A probabilistic solution for the Syracuse conjecture

We prove the veracity of the Syracuse conjecture by establishing that starting from an arbitrary positive integer, the Syracuse process will never reach any integer reached before and then we conclude by using a probabilistic (a random walk) approach.Classification: MSC: 11A25

Game with a random second player and its application to the problem of optimal fare choice

The choice of the optimal strategy for a significant number of applied problems can be formalized as a game theory problem, even in conditions of incomplete information. The article deals with a hierarchical game with a random second player, in which the first player chooses a deterministic solution, and the second player is represented by a set of decision makers. The strategies of the players that ensure the Stackelberg equilibrium are studied. The strategy of the second player is formalized as a probabilistic solution to an optimization problem with an objective function depending on a continuously distributed random parameter. In many cases, the choice of optimal strategies takes place in conditions when there are many decision makers, and each of them chooses a decision based on his (her) criterion. The mathematical formalization of such problems leads to the study of probabilistic solutions to problems with an objective function depending on a random parameter. In particular, probabilistic solutions are used for mathematical describing the passenger's choice of a mode of transport. The problem of optimal fare choice for a new route based on a probabilistic model of passenger preferences is considered. In this formalization, the carrier that sets the fare is treated as the first player; the set of passengers is treated as the second player. The second player's strategy is formalized as a probabilistic solution to an optimization problem with a random objective function. A model example is considered.

First-order linear differential equations whose data are complex random variables: Probabilistic solution and stability analysis via densities

<abstract><p>Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.</p></abstract>